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  mod = %Z,
  nf  = %ZtschirnhausTschirnhaus transform. New pol: %Zgpolcomp (different degrees)S1S2A3S3C(4) = 4E(4) = 2[x]2D(4)A4S4C(5) = 5D(5) = 5:2F(5) = 5:4A5S5C(6) = 6 = 3[x]2D_6(6) = [3]2D(6) = S(3)[x]2A_4(6) = [2^2]3F_18(6) = [3^2]2 = 3 wr 22A_4(6) = [2^3]3 = 2 wr 3S_4(6d) = [2^2]S(3)S_4(6c) = 1/2[2^3]S(3)F_18(6):2 = [1/2.S(3)^2]2F_36(6) = 1/2[S(3)^2]22S_4(6) = [2^3]S(3) = 2 wr S(3)L(6) = PSL(2,5) = A_5(6)F_36(6):2 = [S(3)^2]2 = S(3) wr 2L(6):2 = PGL(2,5) = S_5(6)A6S6C(7) = 7D(7) = 7:2F_21(7) = 7:3F_42(7) = 7:6L(7) = L(3,2)A7S7galoisgalois of reducible polynomialgalois (bug1)galois (bug4)galois (bug2)galois (bug3)galois of degree higher than 11incorrect galois automorphism in galoisapplygaloisapplyget_bnfpolfalse nf in nf_get_r1false nf in nf_get_r2false nf in nf_get_signmatricesmult. tablenfinitround4LLL basispolrednon-monic polynomial. Result of the form [nf,c]incorrect nf in nfnewpreci = %ld
polred for non-monic polynomialordrednf_ADDZK flag when nf_ALL set (polredabs)Found %ld minimal polynomials.
rootsof1rootsof1 (bug1)not an integer type in dirzetaktoo many terms in dirzetakneed %Z coefficients in initzeta: computation impossible
initzeta: N0 = %Z
i0 = %ld
Ciklog(n)a(n)coefa(i,j)not a zeta number field in zetakallgzetakalls = 0 is a pole (gzetakall)s = 1 is a pole (gzetakall)nsiso0nfiso or nfinclmatrix Mget_red_Gget_red_G: starting LLL, prec = %ld (%ld + %ld)
xbest = %Z
you found a counter-example to a conjecture, please report!polredabs0  generator: %Z
get_polpolredabs (precision problem)chk_gen_init: new prec = %ld (initially %ld)
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precision too low in chk_gen_initchk_gen_init: generator %Z
chk_gen_init: difficult field, trying random elements
chk_gen_init: subfield %Z
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%Z
Treating p^k = %Z^%ld
impossible inverse: %Zdisc. factorisationreducible polynomial in allbasenfbasis  dedek: gcd has degree %ld
initial parameters p=%Z,
  f=%Z
  new order: %Z
  entering Dedekind Basis with parameters p=%Z
  f = %Z,
  a = %Z
--> %Z : IndexPartial: factor %Z^%ld IndexPartial: factorizationIndexPartial: discriminantget_normprimedecincorrect modpr formatmodpr initialized for integers only!nf_to_ffincorrect polynomial in rnf functionincorrect coeff in rnf functionrnf functionnon-monic relative polynomialsincorrect variable in rnf functionIdeals to consider:
%Z^%ld
not a pseudo-basis in nfsimplifybasisrnfdet2not a pseudo-matrix in rnfdetrnfsteinitznot a pseudo-matrix in %srnfbasisrnfisfreepolcompositum0compositumnot the same variable in compositumcompositum: %Z inseparablernfequationinseparable relative equation in rnfequationrnflllgram
%ld k = rnfpolredrnfpolredabsreduced absolute generator: %Z
absolute basisoriginal absolute generator: %Z
this combination of flags in rnfpolredabsnot a factorisation in nfbasis  leaving Decomp: f1 = %Z
f2 = %Z
e = %Z
de= %Z
bug in Decomp (not a factor), is p = %Z a prime?  entering Decomp, parameters: %Z^%ld
  f = %Z  (Fa, Ea) = (%ld,%ld)
  entering Nilord with parameters: %Z^%ld
  fx = %Z, gx = %Zprimedec: %Z is not primernfdedekindrnfordmax new order:
%Z
%Z
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lllint[1], kmax = %ldK%ld k = lllfp[1] (%ld) K%ldlllfplllfp[2], kmax = %ldin precision %ld

Checking LLL basis...
Checking LLL basis
lllfp[1], kmax = %ld
Recomputing Gram-Schmidt, kmax = %ld
 %ld
...LLL reducing precision to %ld
k =lllfp giving upcount_max = %ld
lllfp (exact)dependent vectors in lllfpqflllqflllgramgramlllintpartialmid = %Znpass = %ld, red. last time = %ld, log_2(det) ~ %ld

lllintpartialalltm1 = %ZzncoppersmithEntering LLL
bitsize bound: %ld
expected shvector bitsize: %ld
Increasing dim, delta = %d t = %d
Roots: %Z
Candidate: %Z
bitsize Norm: %ld
bitsize bound: %ld
Matrix to be reduced:
%Z
Init: trying delta = %d, t = %d
Modified P: %Z
zero polynomial forbiddennegative bound in zncoppersmithnegative accuracy in lindep2lindep2lindepoverflow in real shiftqzer[%ld]=%ld
pslqInitialization time = %ld
pslqL2inconsistent primes in plindepnot a p-adic vector in plindephigher degree than expected in algdepnegative polynomial degree in algdepalgdep0matkerintqfminimfincke_pohstFincke-Pohst, final LLL: prec = %ld
first LLL: prec = %ld
dimension 0 in fincke_pohstlllgramallgendelta = %d, t = %d, cond = %lf
bound too largetime [max,t12,loop,reds,fin] = [%ld, %ld, %ld, %ld, %ld]
time for ct = %ld : %ld
minim0not a definite form in minim0.minim0, rank>=%ld*negative number of vectors in minim0bound = 0 in minim2maximal number of vectors must be providedfinal sort & check...
sorting...
New bound: %Zsmallvectorssmallvectors looking for norm < %Z
incrementalGSgenadding vector = %Z
vector in new basis = %Z
list = %Z
base change matrix =
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End of PHASE 1.

    #%ld in factor base
    Norm(P) > Zimmert bound
  Testing P = %Z
*** p = %lu
     is %Z
**** Testing Different = %Z
PHASE 1: check primes to Zimmert bound = %lu

red_mod_unitszero ideal in isprincipalisprincipalinsufficient precision for generators, not givenisunitnot a factorization matrix in isunitnot an algebraic number in isunitcodeprimebnfnewprecbnfmakebnfclassunitbnfinitnon-monic polynomial. Change of variables discardedfundamental units too largebnfinit: %s%s, not giveninsufficient precision for fundamental unitsgetfu
#### Computing fundamental units
SPLIT: increasing factor base [%ld]
# ideals tried = %ld
precision too low for generators, not givenprecision too low for generators, e = %ldclassgroup generators
#### Computing class group generators
completing bnf (building cycgen)
*%ld %ld makematalcompleting bnf (building matal)classgroupallincorrect parameters in classgroupbuchall (%s)cleanarchcompute_R
#### Tentative class number: %Z
bestapprregulator is zero.

#### Looking for random relations
LIMC = %ld, LIMC2 = %ld
R1 = %ld, R2 = %ld
D = %Z
Bach constant <= 0 in buchinitalg & rootsof1sub factorbase (%ld elements)*** Changing sub factor base
powFBgen %ldComputing powers for subFB: %Z
factor base########## FACTORBASE ##########

KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld
++ LV[%ld] = %Zisprincipal (incompatible bnf generators)BOUND = %.4g
.small norm relations  small norms gave %ld relations.
  nb. fact./nb. small norm = %ld/%ld = %.3f
for this ideal*v[%ld]=%.4g small_norm (precision too low)
*** Ideal no %ld: [%Z, %Z, %Z, %Z]

#### Looking for %ld relations (small norms)

(more relations needed: %ld)
looking hard for %Z

++++ cglob = %ld: new relation (need %ld)for this relationrelation cancelled: (jid=%ld,jdir=%ld)
m = %Z
(%ld)be_honest() failure on prime %Z
be honestBe honest for %ld primes from %ld to %ld

#### Computing regulator multiple
bestappr/regulator
#### Tentative regulator : %Z

 ***** check = %f
D = %Z
den = %Z
truncation error in bestappr

#### Computing check
KCZ = %ld, KC = %ld, n = %ld
weighted G matricesadding vector = %Z
vector in new basis = %Z
list = %Z
base change =
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PHASE 2: are all primes good ?

  Testing primes <= B (= %lu)

sorry, too many primes to checklarge Minkowski bound: certification will be VERY longMinkowski bound is too largeconductorincorrect subgroup in %srnfnormgroupnon Galois extension in rnfnormgroupnot an Abelian extension in rnfnormgroupnot an Abelian extension in rnfnormgroup?rnfconductorincorrect character length in KerCharconductorofcharbnrclassnolistdiscrayabslistnot a factorisation in decodemoduleincorrect hash code in decodemoduleLbnrclassno[2]: discrayabslistarchStarting discrayabs computations

Starting bnrclassno computations
[1]: discrayabslistarchStarting zidealstarunits computations
r1>15 in discrayabslistarchnon-positive bound in DiscrayabslistsubgrouplistBuchrayMahler bound for regulator: %Z
Default bound for regulator: 0.2
(lower bound for regulator) M = %Z
M0 = %Z
M* = %Z
pol = %Z
old method: y = %Z, M0 = %Z
     p divides h(K)
     Beta list = %Z
     p divides w(K)
  *** testing p = %lu
neither bnf nor bnr in conductor or discraybad subgroup in conductor or discraybnrdiscrayfactordivexact is not exact!*.
Searching minimum of T2-form on units:
   BOUND = %ld
[ %ld, %ld, %ld ]: %Z
       prime ideal Q: %Z
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A4GaloisConj:sigma=%Z 
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S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z
S4GaloisConj:sigma=%Z
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S4GaloisConj:Computing isomorphisms %d:%Z
frobenius powerGaloisConj:next p=%ld
galoisconj _may_ hang up for this polynomialGaloisConj: Fixed field %Z
f=%Z
 borne=%Z
 l-borne=%Z
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%ZM%d.s4test()Best lift: %d
Galoisconj:Subgroups list:%Z
Trying degre %d.
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I will skip it, but it may induce an infinite loopGaloisConj: not found, %d hops 
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currently lost to the factoring machineryMiller-Rabin: testing base %ld
False prime number %Z in plisprimePL: N-1 factored!
PL: proving primality of N = %Z
snextpr: %lu != prc210_rp[%ld] mod 210
[caller of] snextprsnextpr: integer wraparound after prime %lu
snextpr: %lu should have been prime but isn't
ECM: time = %6ld ms,	ellfacteur giving up.
ECM: time = %6ld ms,	p <= %6lu,
	found factor = %Z
	(giant step at p = %lu)
	(baby step table complete)
	(extracted precomputed helix / baby step entries)
ECM: finishing curves %ld...%ld
ECM: time = %6ld ms, entering B2 phase, p = %lu
	(got initial helix)
	(got [p]Q, p = %lu = prc210_rp[%ld] mod 210)
ECM: %lu should have been prime but isn't
ellfacteur	(got [2]Q...[10]Q)
ECM: time = %6ld ms, B1 phase done, p = %lu, setting up for B2
ECM: time = %6ld ms
ECM: dsn = %2ld,	B1 = %4lu,	B2 = %6lu,	gss = %4ld*420
ECM: stack tight, using heap space
ECM: working on %ld curves at a time; initializing for up to %ld rounds...
 for one roundECM: number too small to justify this stage
Rho: searching small factor of %ld-bit integer
Rho: using X^2%+1ld for up to %ld rounds of 32 iterations
Rho: time = %6ld ms,	%3ld rounds, back to normal mode
Rho: fast forward phase (%ld rounds of 64)...
Rho: time = %6ld ms,	Pollard-Brent giving up.
sRho: time = %6ld ms,	%3ld round%s
	Pollard-Brent failed.
composite 	found %sfactor = %Z
	found factors = %Z, %Z,
	and %Z
Rho: hang on a second, we got something here...
	found factor = %Z
Rho: restarting for remaining rounds...
SQUFOF: giving up, time = %ld ms
SQUFOF: second cycle exhausted after %ld iterations,
	dropping it
SQUFOF: found factor %ld^2
SQUFOF: ...but the root form seems to be on the principal cycle
SQUFOF: square form (%ld^2, %ld, %ld) on second cycle
	after %ld iterations, time = %ld ms
SQUFOF: first cycle exhausted after %ld iterations,
	dropping it
SQUFOF: square form (%ld^2, %ld, %ld) on first cycle
	after %ld iterations, time = %ld ms
SQUFOF: blacklisting a = %ld on second cycle
SQUFOF: blacklisting a = %ld on first cycle
SQUFOF: entering main loop with forms
	(1, %ld, %ld) and (1, %ld, %ld)
	of discriminants
	%Z and %Z, respectively
squfof [caller of] (5n is a square)squfof [caller of] (n or 3n is a square)	But it nevertheless wasn't a %ld%s power.
	   %3ld:  %3ld   (3rd %ld, 5th %ld, 7th %ld)
OddPwrs: is %Z
	...a, or 3rd%s, or 5th%s power?
	modulo: resid. (remaining possibilities)
 7th	But it wasn't a pure power.
OddPwrs: passed modular checks
	- ruled out
	checking modulo %ld
OddPwrs: testing for exponent %ld
OddPwrs: examining %Z
ifac_startfactoring 0 in ifac_startIFAC: new partial factorization structure (%ld slots)
...	factor no. %ld is a duplicate%s
 (so far)...	factor no. %ld was unique%s
	stored (largest) factor no. %ld...
IFAC: incorporating set of %ld factor(s)
ifac_decompIFAC: found %ld large prime (power) factor%s.
[2] ifac_decompIFAC: (Partial fact.)Stop requested.
factoring 0 in ifac_decompifac_moebiusifac_issquarefreeifac_omegaifac_bigomegafactor has NULL exponent in ifac_findifac_totientifac_numdivifac_sumdivifac_sumdivk[caller of] elladd0SQUFOF: found factor %ld from ambiguous form
	after %ld steps on the ambiguous cycle, time = %ld ms
SQUFOF: squfof_ambig returned %ld
SQUFOF: ...found nothing on the ambiguous cycle
	after %ld steps there, time = %ld ms
IFAC: main loop: repeated old factor
	%Z
non-existent factor class in ifac_mainIFAC: main loop: another factor was divisible by
	%Z
IFAC: main loop: repeated new factor
	%Z
IFAC: after main loop: repeated old factor
	%Z
IFAC: main loop: %ld factor%s left
IFAC: main loop: this was the last factor
LucasModIFAC: found %Z =
	%Z ^2
IFAC: found %Z =
	%Z ^%ld
ifac_crack [Z_issquarerem miss]IFAC: cofactor = %Z
IFAC: factoring %Z
	yielded `factor' %Z
	which isn't!
factoringIFAC: forcing ECM, may take some time
IFAC: unfactored composite declared prime	%Z
IFAC: untested integer declared primeIFAC: trying MPQS
IFAC: trying Lenstra-Montgomery ECM
IFAC: trying Shanks' SQUFOF, will fail silently if input
      is too large for it.
IFAC: trying Pollard-Brent rho method
IFAC: factor %Z
	is prime
IFAC: checking for odd power
IFAC: checking for pure square
IFAC: cracking composite
	%Z
IFAC: prime %Z
	appears at least to the power %ld
IFAC: a factor was divisible by another prime factor,
	leaving a cofactor = %Z
IFAC: a factor was a power of another prime factor
IFAC: prime %Z
	appears with exponent = %ld
ifac_sort_one`*where' out of bounds in ifac_sort_one`washere' out of bounds in ifac_sort_onemisaligned partial detected in ifac_sort_oneprime equals composite in ifac_sort_onecomposite equals prime in ifac_sort_oneIFAC: repeated factor %Z
	detected in ifac_sort_one
partial impossibly short in ifac_sort_oneIFAC: factor %Z
	is prime (no larger composite)
compositeprimeIFAC: factor %Z
	is %s
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coeff bound: %Z^%ld
Knapsacklast factor still to be checked
Mignotte bound: %Z
Beauzamy bound: %Z
Naive recombinationHensel lift (mod %Z^%ld)RecombinationDDF_roots, m = %ldRoot boundcan't deflate this power series (d = %ld): %Zgdeflateneed positive degree in gdeflatefactpolnfrootsQpartial factorization is not meaningful herecan't factor %Zfactor for general polynomialsfactor of general polynomialleftright_powprod: remaining objects %ld
not a factorisation in factorbackmissing nf in factorbackeltgisirreduciblegcdgRgX_gcd_simplecontentQ_contentQ_denomQ_muli_to_intQ_divmuli_to_intQ_div_to_intmissing case in gdivexactpseudodiv dx = %ld >= %ldsubresall, dr = %ldsubresallsubresext, dr = %ldinexact computation in subresextsubresextRgX_extgcd, dr = %ldinexact computation in RgX_extgcdresultantducos, degpol Q = %ldsylvestermatrixnot the same variables in sylvestermatrixpolresultantsrgcdsrgcd: dr = %ld
discsrreduceddiscsmithpoldiscreducednon-monic polynomial in poldiscreducedsturmpolsturm, dr = %ldnot a squarefree polynomial in sturmginvmodnewtonpolymatratliftpolratliftnfgcdnfgcd: p=%d
lifting to prec %ldbuilding treeMultiLift: bad args
### K = %d, %Z combinations
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found factor %Z
remaining modular factor(s): %ld

*LLL_cmbfLLL_cmbf: chk_factors failed* Time LLL: %ld
* Time Check Factor: %ld
LLL_cmbf: rank decrease
for this block of tracesLLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld

LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)
DDF: wrong numbers of factorsTime setup: %ld
Total Time: %ld
===========
splitting mod p = %ldfactorsroots...tried prime %3ld (%-3ld %s). Time = %ld
pseudorem dx = %ld >= %ldnextSousResultant j = %ld/%ldnon-invertible polynomial in RgXQ_invrelatively prime polynomials expectedLLL_cmbf: checking factor %ld
special_pivot output:
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	d		Rg_to_FpRg_to_FlRg_to_FpXQnon invertible polynomial in FpXQ_invFpX_FpXQV_compo: %d FpXQ_mul [%d]
powers is only [] or [1] in FpX_FpXQV_compoFpXQX_gcdnon-invertible polynomial in FpXQX_gcdFpXQ_sqrtlFpXQ_sqrtnfflgen1/0 exponent in FpXQ_sqrtnbad degrees in FpX_ffintersect: %d,%d,%dFpM_invimagePolynomials not irreducible in FpX_ffintersectpows [P,Q]ZZ_%Z[%Z]/(%Z) is not a field in FpX_ffintersectFp_sqrtnFpM_kerfactor_irredfactor_irred_matFpXQ_matrix_powsFpX_resultant (da = %ld)polint_triv2 (i = %ld)FpV_polintresultant mod %ld (bound 2^%ld, stable=%ld)ZY_ZXY_rnfequationFinal lambda = %ld
Degree list for ERS (trials: %ld) = %Z
bound for resultant coeffs: 2^%ld
Trying lambda = %ld
ZY_ZXY_resultant_all: LERS needs lambdaresultant mod %ld (bound 2^%ld, stable = %d)ZX_resultantbound for resultant: 2^%ld
modulargcddifferent variables in modulargcdgcd mod %lu (bound 2^%ld)bound 2^%ld. Goal 2^%ldmodulargcd: trial division failedQXQ_invQXQ_inv: mod %ld (bound 2^%ld)QXQ_inv: char 0 check failedffinitFFInit: using subcyclo(%ld, %ld)
non positive degree in ffinitFF l-Gen:next %Z
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I9OO(a^b)=o(a^b)=p-adic or power series zero with precision given by babsGpabs(x)=absolute value (or modulus) of xacosacos(x)=inverse cosine of xacoshacosh(x)=inverse hyperbolic cosine of xaddellGGGaddell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve eaddprimesGaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableadjadj(x)=adjoint matrix of xagmGGpagm(x,y)=arithmetic-geometric mean of x and yakellGGakell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve ealgdepGLpalgdep(x,n)=algebraic relations up to degree n of xalgdep2GLLpalgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in  lindep2algtobasisalgtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]anellGLanell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)apellapell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodapell2apell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsapprpadicapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod pargarg(x)=argument of x,such that -pi<arg(x)<=piasinasin(x)=inverse sine of xasinhasinh(x)=inverse hyperbolic sine of xassmatassmat(x)=associated matrix to polynomial xatanatan(x)=inverse tangent of xatanhatanh(x)=inverse hyperbolic tangent of xbasisGfbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithmbasis2basis2(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 2 algorithmbasistoalgbasistoalg(nf,x)=transforms the vertical vector x on the integral basis into an algebraic numberbernrealLpbernreal(x)=Bernoulli number B_x, as a real number with the current precisionbernvecLbernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)bestapprbestappr(x,k)=gives the best approximation to the real x with denominator less or equal to kbezoutbezout(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbezoutresbezoutres(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbigomegabigomega(x)=number of repeated prime divisors of xbilhellGGGpbilhell(e,z1,z2)=canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsbinbin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbinarybinary(x)=gives the vector formed by the binary digits of x (x C-integer)bittestbittest(x,n)=gives bit number n (coefficient of 2^n) of the integer xboundcfboundcf(x,lmax)=continued fraction expansion of x with at most lmax termsboundfactboundfact(x,lim)=partial factorization of the integer x (using primes up to lim)buchcertifylGbuchcertify(bnf)=certify the correctness (i.e. remove the GRH) of the bnf data output by buchinit or buchinitfubuchfubuchfu(bnf)=compute the fundamental units of the number field bnf output by buchinitbuchgenGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,pbuchgen(P,...)=compute the structure of the class group and the regulator for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchgenforcefubuchgenforcefu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units for the number field defined by the polynomial P, and insist until the units are obtained. See manual for the other parameters (which can be omitted)buchgenfubuchgenfu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units (if they are not too large) for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchimagGD0.1,G,D0.1,G,D5,G,buchimag(D,...)=compute the structure of the class group of the complex quadratic field of discriminant D<0. See manual for the other parameters (which can be omitted)buchinitbuchinit(P,...)=compute the necessary data for future use in ideal and unit group computations. See manual for detailsbuchinitforcefubuchinitforcefu(P,...)=compute the necessary data for future use in ideal and unit group computations, and insist on having fundamental units. See manual for detailsbuchinitfubuchinitfu(P,...)=compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. See manual for detailsbuchnarrowbuchnarrow(bnf)=given a big number field as output by buchinitxx, gives as a 3-component vector the structure of the narrow class groupbuchraybuchray(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this modulebuchrayinitbuchrayinit(bnf,ideal)=same as buchrayinitgen, except that the generators are not explicitly computedbuchrayinitgenbuchrayinitgen(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data for computing in the ray class group  corresponding to this module. In particular, the fifth component is the ray class group structurebuchrealGD0,G,D0.1,G,D0.1,G,D5,G,pbuchreal(D,...)=compute the structure of the class group and the regulator of the real quadratic field of discriminant D>0 in the wide sense. See manual for the other parameters (which can be omitted)bytesizebytesize(x)=number of bytes occupied by the complete tree of the object xceilceil(x)=ceiling of x=smallest integer>=xcenterliftcenterlift(x)=centered lift of x. Same as lift except for integermodscfcf(x)=continued fraction expansion of x (x rational,real or rational function)cf2cf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchangevarchangevar(x,y)=change variables of x according to the vector ycharGnchar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchar1char1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchar2char2(x,y)=characteristic polynomial of the matrix x expressed with variable y, using the Hessenberg form. Can be much faster or much slower than char, depending on the base ringchellchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]chinesechinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptellchptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]classnoclassno(x)=class number of discriminant xclassno2classno2(x)=class number of discriminant xcoeffcoeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[])compimagcompimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompocompo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]compositumcompositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2compositum2compositum2(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2, with roots of pol1 and pol2 expressed on the compositum polynomialscomprealrawcomprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantconcatconcat(x,y)=concatenation of x and yconductorGDGDGD1,G,conductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupconductorofcharconductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrconjconj(x)=the algebraic conjugate of xconjvecconjvec(x)=conjugate vector of the algebraic number xcontentcontent(x)=gcd of all the components of x, when this makes senseconvolconvol(x,y)=convolution (or Hadamard product) of two power seriescorecore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarecore2core2(n)=(long)gen_2-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisccoredisc(n)=discriminant of the quadratic field Q(sqrt(n))coredisc2coredisc2(n)=(long)gen_2-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercoscos(x)=cosine of xcoshcosh(x)=hyperbolic cosine of xcvtoicvtoi(x)=truncation of x, without taking into account loss of integer part precisioncycloLDncyclo(n)=n-th cyclotomic polynomialdecodefactordecodefactor(fa)=given a factorisation fa, gives the factored object backdecodemoduledecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduledegreedegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardenomdenom(x)=denominator of x (or lowest common denominator in case of an array)deplindeplin(x)=finds a linear dependence between the columns of the matrix xderivderiv(x,y)=derivative of x with respect to the main variable of ydetdet(x)=determinant of the matrix xdet2det2(x)=determinant of the matrix x (better for integer entries)detintdetint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemoddiagonaldiagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdilogdilog(x)=dilogarithm of xdirdivdirdiv(x,y)=division of the Dirichlet series x by the Dir. series ydireulerV=GGIDGdireuler(p=a,b,expr)=Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s)dirmuldirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ydirzetakdirzetak(nf,b)=Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1discdisc(x)=discriminant of the polynomial xdiscfdiscf(x)=discriminant of the number field defined by the polynomial x using round 4discf2discf2(x)=discriminant of the number field defined by the polynomial x using round 2discrayabsGD0,G,D0,G,D0,L,discrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayabscondGD0,G,D0,G,D2,L,discrayabscond(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if fmodule is not the conductordiscrayabslistdiscrayabslist(bnf,listes)=if listes is a 2-component vector as output by ideallistunit or similar, gives list of corresponding discrayabsconddiscrayabslistarchGGLdiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formatdiscrayabslistarchalldiscrayabslistarchall(bnf,bound)=gives list of discrayabscond of all modules up to norm bound with all possible archimedean places arch in reverse lexicographic order, in a longvector formatdiscrayabslistlongdiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatdiscrayrelGD0,G,D0,G,D1,L,discrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayrelcondGD0,G,D0,G,D3,L,discrayrelcond(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if module is not the conductordivisorsdivisors(x)=gives a vector formed by the divisors of x in increasing orderdivresdivres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderdivsumGVIdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neigeneigen(x)=eigenvectors of the matrix x given as columns of a matrixeint1eint1(x)=exponential integral E1(x)erfcerfc(x)=complementary error functionetaeta(x)=eta function without the q^(1/24)eulerpeuler=euler()=euler's constant with current precisionevaleval(x)=evaluation of x, replacing variables by their valueexpexp(x)=exponential of xextractextract(x,y)=extraction of the components of the vector x according to the vector or mask y, from left to right (1, 2, 4, 8, ...for the first, second, third, fourth,...component)factfact(x)=factorial of x (x C-integer), the result being given as a real numberfactcantorfactcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactfqfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factmodfactmod(x,p)=factorization mod p of the polynomial x using Berlekampfactorfactor(x)=factorization of xfactoredbasisGGffactoredbasis(x,p)=integral basis of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoreddiscffactoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoredpolredfactoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives minimal polynomials only)factoredpolred2factoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives elements and minimal polynomials)factornffactornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial tfactorpadicfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmfactorpadic2factorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-LenstrafactpolGLLfactpol(x,l,hint)=factorization over Z of the polynomial x up to degree l (complete if l=0) using Hensel lift, knowing that the degree of each factor is a multiple of hintfactpol2factpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingfibofibo(x)=fibonacci number of index x (x C-integer)floorfloor(x)=floor of x=largest integer<=xforvV=GGIfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bfordivvGVIfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of nforprimeforprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bforstepvV=GGGIforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sforvecvV=GID0,L,forvec(x=v,seq)=v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1fpnGLDnfpn(p,n)=monic irreducible polynomial of degree n over F_p[x]fracfrac(x)=fractional part of x=x-floor(x)galoisgalois(x)=Galois group of the polynomial x (see manual for group coding)galoisapplygaloisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconjgaloisconj(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field, using p-adics, LLL on integral basis (not always complete)galoisconj1galoisconj1(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field nf, using complex numbers, LLL on integral basis (not always complete)galoisconjforcegaloisconjforce(nf)=list of conjugates of a root of the polynomial x=nf[1] in the Galois number field nf, using p-adics, LLL on integral basis. Guaranteed to be complete if the field is Galois, otherwise there is an infinite loopgamhgamh(x)=gamma of x+1/2 (x integer)gammagamma(x)=gamma function at xgaussgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)gaussmodulogaussmodulo(M,D,Y)=(long)gen_1 solution of system of congruences MX=Y mod Dgaussmodulo2gaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dgcdgcd(x,y)=greatest common divisor of x and ygetheapgetheap()=2-component vector giving the current number of objects in the heap and the space they occupygetrandlgetrand()=current value of random number seedgetstackgetstack()=current value of stack pointer avmagettimegettime()=time (in milliseconds) since last call to gettimeglobalredglobalred(e)=e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pgotos*goto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhclassnohclassno(x)=Hurwitz-Kronecker class number of x>0hellhell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshell2hell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermitehermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmhermite2hermite2(x)=2-component vector [H,U] such that H is an (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, and U is a unimodular matrix such that xU=H, using Batut's algorithmhermitehavashermitehavas(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Havas's algorithmhermitemodhermitemod(x,d)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is the non-zero determinant of this latticehermitemodidhermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixhermitepermhermiteperm(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Batut's algorithmhesshess(x)=Hessenberg form of xhilblGGGhilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)hilberthilbert(n)=Hilbert matrix of order n (n C-integer)hilbplGGhilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hvectorhvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nhyperuhyperu(a,b,x)=U-confluent hypergeometric functionii=i()=square root of -1idealaddidealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddmultoneidealaddone(nf,x,y)=when the sum of two ideals x and y in the number field K defined by nf is equal to Z_K, gives a two-component vector [a,b] such that a is in x, b is in y and a+b=1idealaddoneidealaddmultone(nf,list)=when the sum of the ideals in the number field K defined by nf and given in the vector list is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1idealappridealappr(nf,x)=x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealapprfactidealapprfact(nf,x)=x being a prime ideal factorization with possibly zero or negative exponents, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealchineseidealchinese(nf,x,y)=x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprimeidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yidealdividealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealdivexactidealdivexact(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nf when the quotient is known to be an integral idealidealfactoridealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermiteidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveidealhermite2idealhermite2(nf,a,b)=hermite normal form of the ideal aZ_K+bZ_K in the number field K defined by nf, where a and b are elementsidealintersectidealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfidealinvidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentidealinv2idealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentideallistideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfideallistarchideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsideallistarchgenideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsideallistunitideallistunit(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, without generatorsideallistunitarchideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitideallistunitarchgenideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallistunitgenideallistunitgen(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, with generatorsideallistzstarideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsideallistzstargenideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsideallllredideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealmulidealmul(nf,x,y)=product of the two ideals x and y in the number field nfidealmulredidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealnormidealnorm(nf,x)=norm of the ideal x in the number field nfidealpowidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfidealpowredidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfidealtwoeltidealtwoelt(nf,x)=(long)gen_2-element representation of an ideal x in the number field nfidealtwoelt2idealtwoelt2(nf,x,a)=(long)gen_2-element representation of an ideal x in the number field nf, with the first element equal to aidealvalidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfidmatidmat(n)=identity matrix of order n (n C-integer)ifif(a,seq1,seq2)=if a is nonzero, seq1 is evaluated, otherwise seq2imagimag(x)=imaginary part of ximageimage(x)=basis of the image of the matrix ximage2image2(x)=basis of the image of the matrix ximagecomplimagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgamincgam(s,x)=incomplete gamma functionincgam1incgam1(s,x)=incomplete gamma function (for debugging only)incgam2incgam2(s,x)=incomplete gamma function (for debugging only)incgam3incgam3(s,x)=complementary incomplete gamma functionincgam4incgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedindexrankindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankindsortindsort(x)=indirect sorting of the vector xinitalginitalg(x)=x being a nonconstant irreducible polynomial, gives the vector: [x,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgredinitalgred(x)=x being a nonconstant irreducible polynomial, finds (using polred) a simpler polynomial pol defining the same number field, and gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual), r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgred2initalgred2(P)=P being a nonconstant irreducible polynomial, gives a two-element vector [nf,mod(a,pol)], where nf is as output by initalgred and mod(a,pol) is a polymod equal to mod(x,P) and pol=nf[1]initellinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]initzetainitzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialinteginteg(x,y)=formal integration of x with respect to the main variable of yintersectintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yintgenV=GGID1,L,pintgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularitiesintinfV=GGID2,L,pintinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same signintnumV=GGID0,L,pintnum(X=a,b,s)=numerical integration of s from a to b with respect to XintopenV=GGID3,L,pintopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or binverseimageinverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonalisdiagonal(x)=true(1) if x is a diagonal matrix, false(0) otherwiseisfundisfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisidealisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisinclisincl(x,y)=tests whether the number field defined by the polynomial x is isomorphic to a subfield of the one defined by y; 0 if not, otherwise all the isomorphismsisinclfastisinclfast(nf1,nf2)=tests whether the number nf1 is isomorphic to a subfield of nf2 or not. If it gives a non-zero result, this proves that this is the case. However if it gives zero, nf1 may still be isomorphic to a subfield of nf2 so you have to use the much slower isincl to be sureisirreducibleisirreducible(x)=true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantisisomisisom(x,y)=tests whether the number field defined by the polynomial x is isomorphic to the one defined by y; 0 if not, otherwise all the isomorphismsisisomfastisisomfast(nf1,nf2)=tests whether the number fields nf1 and nf2 are isomorphic or not. If it gives a non-zero result, this proves that they are isomorphic. However if it gives zero, nf1 and nf2 may still be isomorphic so you have to use the much slower isisom to be sureisoncurveiGGisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisprimeGD0,L,isprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notisprincipalisprincipal(bnf,x)=bnf being output by buchinit, gives the vector of exponents on the class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalforceisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedisprincipalgenisprincipalgen(bnf,x)=bnf being output by buchinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorisprincipalgenforceisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedisprincipalrayisprincipalray(bnf,x)=bnf being output by buchrayinit, gives the vector of exponents on the ray class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalraygenisprincipalraygen(bnf,x)=bnf being output by buchrayinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorispspispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrtisqrt(x)=integer square root of x (x integer)issetisset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notissqfreeissqfree(x)=true(1) if x is squarefree, false(0) if notissquareissquare(x)=true(1) if x is a square, false(0) if notisunitisunit(bnf,x)=bnf being output by buchinit, gives the vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisejacobijacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xjbesselhjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerjelljell(x)=elliptic j invariant of xkaramulkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDkbesselkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kbessel2kbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kerker(x)=basis of the kernel of the matrix xkerikeri(x)=basis of the kernel of the matrix x with integer entrieskerintkerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint1kerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFkerint2kerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkrokro(x,y)=kronecker symbol (x/y)labellabel(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlambdaklambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)laplacelaplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^nlcmlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)legendrelegendre(n)=legendre polynomial of degree n (n C-integer)lengthlength(x)=number of non code words in xlexlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lexsortlexsort(x)=sort the elements of the vector x in ascending lexicographic orderliftlift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]lindeplindep(x)=Z-linear dependencies between components of x (Hastad et al)lindep2lindep2(x,dec)=Z-linear dependencies between components of x using LLL, where dec should be about one half the number of decimal digits of precisionllllll(x)=lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lll1lll1(x)=old version of lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgenlllgen(x)=lll reduction of the vectors forming the matrix x with polynomial coefficients (gives the unimodular transformation matrix)lllgramlllgram(x)=lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgram1lllgram1(x)=old version of lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgramgenlllgramgen(x)=lll reduction of the lattice whose gram matrix is x with polynomial coefficients (gives the unimodular transformation matrix)lllgramintlllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x (gives the unimodular transformation matrix)lllgramkerimlllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is the integral matrix xlllgramkerimgenlllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is the matrix x with polynomial coefficientslllintlllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllintpartiallllintpartial(x)=partial (hence faster) lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllkerimlllkerim(x)=kernel and lll reduction of the vectors forming the integral matrix xlllkerimgenlllkerimgen(x)=kernel and lll reduction of the vectors forming the matrix x with polynomial coefficientslllratlllrat(x)=lll reduction of the vectors forming the matrix x, computations done with rational numbers (gives the unimodular transformation matrix)lnln(x)=log(x)=natural logarithm of xlngammalngamma(x)=logarithm of the gamma function of xlocalredlocalred(e,p)=e being an ellliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_ploglog(x)=ln(x)=natural logarithm of xlogagmlogagm(x)=natural logarithm of x, computed using agm (faster than log for more than a few hundred decimal digits)lseriesellGGGGplseriesell(e,s,N,A)=L-series at s of the elliptic curve e, where |N| is the conductor, sign(N) the sign of the functional equation, and A a cut-off point close to 1makebigbnfmakebigbnf(sbnf)=transforms small sbnf as output by smallbuchinit into a true big bnfmatmat(x)=transforms any GEN x into a matrixmatextractmatextract(x,y,z)=extraction of the components of the matrix x according to the vector or masks y (for the rows) and z (for the columns) from left to right (1,2,4,8,...for the first, second, third, fourth, ...rows or columns)mathellmathell(e,x)=gives the height matrix for vector of points x on elliptic curve e using theta functionsmatrixGGVVImatrix(m,n,X,Y,expr)=mXn matrix of expression expr, the row variable X going  from 1 to m and the column variable Y going from 1 to nmatrixqzmatrixqz(x,p)=transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwisematrixqz2matrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xmatrixqz3matrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsizematsize(x)=number of rows and columns of the vector/matrix x as a 2-vectormaxmax(x,y)=maximum of x and yminmin(x,y)=minimum of x and yminidealminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfminimminim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0minim2minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normmodmod(x,y)=creates the integer x modulo y on the PARI stackmodpmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)modreversemodreverse(x)=reverse polymod of the polymod x, if it existsmodulargcdmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodmumu(x)=Moebius function of xnewtonpolynewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnextprimenextprime(x)=smallest prime number>=xnfdetintnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfdivnfdiv(nf,a,b)=element a/b in nfnfdiveucnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnfdivresnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfhermitenfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhermitemodnfhermitemod(nf,x,detx)=if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfmodnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernfmulnfmul(nf,a,b)=element a.b in nfnfpownfpow(nf,a,k)=element a^k in nfnfreducenfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallnfsmithnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfvalnfval(nf,a,pr)=valuation of element a at the prime prnormnorm(x)=norm of xnorml2norml2(x)=square of the L2-norm of the vector xnucompnucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputednumdivnumdiv(x)=number of divisors of xnumernumer(x)=numerator of xnupownupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduploo(a^b)=O(a^b)=p-adic or power series zero with precision given by bomegaomega(x)=number of unrepeated prime divisors of xordellordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve eorderorder(x)=order of the integermod x in (Z/nZ)*orderellorderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionordredordred(x)=reduction of the polynomial x, staying in the same orderpadicprecpadicprec(x,p)=absolute p-adic precision of object xpascalLDGpascal(n)=pascal triangle of order n (n C-integer)perfperf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apermutationLGpermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)permutation2numpermutation2num(vect)=ordinal (between 1 and n!) of permutation vectpfpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xphiphi(x)=Euler's totient function of xpipi=pi()=the constant pi, with current precisionpnqnpnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpointellpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zpolintGGGD&polint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolredpolred(x)=reduction of the polynomial x (gives minimal polynomials only)polred2polred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabspolredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normpolredabs2polredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polpolredabsallpolredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolredabsfastpolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolredabsnoredpolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolsympolsym(x,n)=vector of symmetric powers of the roots of x up to npolvarpolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarspolypoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientpolylogLGppolylog(m,x)=m-th polylogarithm of xpolylogdpolylogd(m,x)=D_m~-modified m-th polylog of xpolylogdoldpolylogdold(m,x)=D_m-modified m-th polylog of xpolylogppolylogp(m,x)=P_m-modified m-th polylog of xpolyrevpolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termpolzagLLpolzag(n,m)=Zagier's polynomials of index n,mpowellpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)powrealrawpowrealraw(x,n)=n-th power without reduction of the binary quadratic form x of positive discriminantprecprec(x,n)=change the precision of x to be n (n C-integer)precisionprecision(x)=real precision of object xprimeprime(n)=returns the n-th prime (n C-integer)primedecprimedec(nf,p)=prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a.Z_K, e,f as usual, a as vector of components on the  integral basis, b Lenstra's constantprimesprimes(n)=returns the vector of the first n primes (n C-integer)primrootprimroot(n)=returns a primitive root of n when it existsprincipalidealprincipalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfprincipalideleprincipalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfprodGV=GGIprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionprodeulerV=GGIpprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionprodinfV=GID0,L,pprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionprodinf1V=GID1,L,pprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionpsipsi(x)=psi-function at xqfiqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0qfrGGGGqfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance dquaddiscquaddisc(x)=discriminant of the quadratic field Q(sqrt(x))quadgenquadgen(x)=standard generator of quadratic order of discriminant xquadpolyquadpoly(x)=quadratic polynomial corresponding to the discriminant xrandomDGrandom()=random integer between 0 and 2^31-1rankrank(x)=rank of the matrix xrayclassnorayclassno(bnf,x)=ray class number of the module x for the big number field bnf. Faster than buchray if only the ray class number is wantedrayclassnolistrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersrealreal(x)=real part of xreciprecip(x)=reciprocal polynomial of xredimagredimag(x)=reduction of the binary quadratic form x with D<0redrealredreal(x)=reduction of the binary quadratic form x with D>0redrealnodredrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]reduceddiscreduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fregularegula(x)=regulator of the real quadratic field of discriminant xreorderreorder(x)=reorder the variables for output according to the vector xresultantresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2resultant2(x,y)=resultant of the polynomials x and yreversereverse(x)=reversion of the power series xrhorealrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrhorealnodrhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rndtoirndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfbasisrnfbasis(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfdiscfrnfdiscf(nf,pol)=given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfequationrnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polrnfequation2rnfequation2(nf,pol)=given a pol with coefficients in nf, gives [apol,th], where apol is the absolute equation of the number field defined by pol and th expresses the root of nf[1] in terms of the root of apolrnfhermitebasisrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisfreernfisfree(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnflllgramrnflllgram(nf,pol,order)=given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfpolredrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplerrnfpseudobasisrnfpseudobasis(nf,pol)=given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitzrnfsteinitz(nf,order)=given an order as output by rnfpseudobasis, gives [A,I,..] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialrootmodrootmod(x,p)=roots mod p of the polynomial xrootmod2rootmod2(x,p)=roots mod p of the polynomial x, when p is smallrootpadicrootpadic(x,p,r)=p-adic roots of the polynomial x to precision rrootsroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonrootsof1rootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfrootsoldrootsold(x)=roots of the polynomial x using a modified Newton's methodroundround(x)=take the nearest integer to all the coefficients of xrounderrorrounderror(x)=maximum error found in rounding xseriesseries(x,v)=convert x (usually a vector) into a power series with variable v, starting with the constant coefficientsetset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientssetintersectsetintersect(x,y)=intersection of the sets x and ysetminussetminus(x,y)=set of elements of x not belonging to ysetrandlLsetrand(n)=reset the seed of the random number generator to nsetsearchlGGD0,L,setsearch(x,y)=looks if y belongs to the set x. Returns 0 if it is not, otherwise returns the index j such that y==x[j]setunionsetunion(x,y)=union of the sets x and yshiftshift(x,n)=shift x left n bits if n>=0, right -n bits if n<0shiftmulshiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)sigmasigma(x)=sum of the divisors of xsigmaksigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)signiGsign(x)=sign of x, of type integer, real or fractionsignatsignat(x)=signature of the symmetric matrix xsignunitsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsimplefactmodsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensimplifysimplify(x)=simplify the object x as much as possiblesinsin(x)=sine of xsinhsinh(x)=hyperbolic sine of xsizesize(x)=maximum number of decimal digits minus one of (the coefficients of) xsmallbasissmallbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallbuchinitsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsmalldiscfsmalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallfactsmallfact(x)=partial factorization of the integer x (using only the stored primes)smallinitellsmallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]smallpolredsmallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)smallpolred2smallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)smithsmith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsmith2smith2(x)=gives a three element vector [u,v,d] where u and v are square unimodular matrices such that d=u*x*v=diagonal(smith(x))smithcleansmithclean(z)=if z=[u,v,d] as output by smith2, removes from u,v,d the rows and columns corresponding to entries equal to 1 in dsmithpolsmithpol(x)=Smith normal form (i.e. elementary divisors) of the matrix x with polynomial coefficients, expressed as a vectorsolvesolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sortsort(x)=sort in ascending order of the vector xsqrsqr(x)=square of x. NOT identical to x*xsqredsqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)sqrtsqrt(x)=square root of xsrgcdsrgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmsturmsturm(x)=number of real roots of the polynomial xsturmpartsturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]subcycloLLDnsubcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powersubellsubell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esubstGnGsubst(x,y,z)=in expression x, replace the variable y by the expression zsumsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprsumaltsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at asumalt2sumalt2(X=a,expr)=Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at asuminfV=GIpsuminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression exprsumpossumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at asumpos2sumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialssupplementsupplement(x)=supplement the columns of the matrix x to an invertible matrixsylvestermatrixsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstantan(x)=tangent of xtanhtanh(x)=hyperbolic tangent of xtaniyamaGPtaniyama(e)=modular parametrization of elliptic curve etaylorGnPtaylor(x,y)=taylor expansion of x with respect to the main variable of ytchebitchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)teichteich(x)=teichmuller character of p-adic number xthetatheta(q,z)=Jacobi sine theta-functionthetanullkthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)threetotwothreetotwo(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]threetotwo2threetotwo2(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]torselltorsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorstracetrace(x)=trace of xtranstrans(x)=x~=transpose of xtrunctrunc(x)=truncation of x;when x is a power series,take away the O(X^)tschirnhaustschirnhaus(x)=random Tschirnhausen transformation of the polynomial xtwototwotwototwo(nf,a,b)=returns a 3-component vector [d,e,U] such that U is a unimodular 2x2 matrix with algebraic integer coefficients such that [a,b]*U=[d,e] and d,e are hopefully smallerunitunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positiveuntiluntil(a,seq)=evaluate the expression sequence seq until a is nonzerovaluationvaluation(x,p)=valuation of x with respect to pvecvec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecindexsortvecindexsort(x): indirect sorting of the vector xveclexsortveclexsort(x): sort the elements of the vector x in ascending lexicographic ordervecmaxvecmax(x)=maximum of the elements of the vector/matrix xvecminvecmin(x)=minimum of the elements of the vector/matrix xvecsortvecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentvectorvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)vvectorvvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)weipellweipell(e)=formal expansion in x=z of Weierstrass P functionwfweberf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)wf2weberf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)whilewhile(a,seq)=while a is nonzero evaluate the expression sequence seq. Otherwise 0zellzell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezetazeta(s)=Riemann zeta function at szetakzetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)zideallogzideallog(nf,x,bid)=if bid is a big ideal as given by zidealstarinit or zidealstarinitgen , gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)zidealstarzidealstar(nf,I)=3-component vector v, giving the structure of (Z_K/I)^*. v[1] is  the order (i.e. phi(I)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorszidealstarinitzidealstarinit(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*zidealstarinitgenzidealstarinitgen(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*znstarznstar(n)=3-component vector v, giving the structure of (Z/nZ)^*. v[1] is  the order (i.e. phi(n)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorsthis function has been suppressedyPD?k@⍀@PWR+⍀+P>9⍀P% ⍀P⍀P⍀P⍀P⍀P⍀P⍀Pvqn⍀nP]XY⍀YPD?D⍀DP+::::;;3;L;e;~;;;;$Ë$h@8,GG
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ColDGCol({x=[]}): transforms the object x into a column vector. Empty vector if x is omittedEulerpEuler=Euler(): Euler's constant with current precisionII=I(): square root of -1ListList({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedMatMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedModGGD0,L,Mod(x,y): creates 'x modulo y'.OO(a^b): p-adic or power series zero with precision given by bPiPi=Pi(): the constant pi, with current precisionPolGDnPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientPolrevPolrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termQfbGGGDGpQfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0SerSer(x,{v=x}): convert x (usually a vector) into a power series with variable v, starting with the constant coefficientSetSet({x=[]}): convert x into a set, i.e. a row vector with strictly increasing coefficients. Empty set if x is omittedStrs*Str({str}*): concatenates its (string) argument into a single stringStrchrGStrchr(x): converts x to a string, translating each integer into a characterStrexpandStrexpand({str}*): concatenates its (string) argument into a single string, performing tilde expansionStrtexStrtex({str}*): translates its (string) arguments to TeX format and returns the resulting stringVecVec({x=[]}): transforms the object x into a vector. Empty vector if x is omittedVecsmallVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedabsGpabs(x): absolute value (or modulus) of xacosacos(x): inverse cosine of xacoshacosh(x): inverse hyperbolic cosine of xaddhelpvSsaddhelp(symbol,"message"): add/change help message for a symboladdprimesaddprimes({x=[]}): add primes in the vector x to the prime table to be used in trial division. x may also be a single integer. Composite "primes" are allowed, and in that case you may later get a message "impossible inverse", which will give you some factors. List the current extra primes if x is omitted. If some primes are added which intersect non trivially the existing table entries, suitable updating is doneagmGGpagm(x,y): arithmetic-geometric mean of x and yalgdepGLD0,L,palgdep(x,n,{flag=0}): algebraic relations up to degree n of x, using lindep([1,x,...,x^(n-1)], flag).aliasvrralias("new","old"): new is now an alias for oldallocatememvD0,L,allocatemem({s=0}): allocates a new stack of s bytes. doubles the stack if s is omittedargarg(x): argument of x,such that -pi<arg(x)<=piasinasin(x): inverse sine of xasinhasinh(x): inverse hyperbolic sine of xatanatan(x): inverse tangent of xatanhatanh(x): inverse hyperbolic tangent of xbernfracLbernfrac(x): Bernoulli number B_x, as a rational numberbernrealLpbernreal(x): Bernoulli number B_x, as a real number with the current precisionbernvecbernvec(x): Vector of rational Bernoulli numbers B_0, B_2,...up to B_(2x)besselh1besselh1(nu,x): H^1-bessel function of index nu and argument xbesselh2besselh2(nu,x): H^2-bessel function of index nu and argument xbesselibesseli(nu,x): I-bessel function of index nu and argument xbesseljbesselj(nu,x): J-bessel function of index nu and argument xbesseljhbesseljh(n,x): J-bessel function of index n+1/2 and argument x, where n is a non-negative integerbesselkGGD0,L,pbesselk(nu,x,{flag=0}): K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type). flag is optional, and may be set to 0: default; 1: use hyperubesselnbesseln(nu,x): N-bessel function of index nu and argument xbestapprGGDGbestappr(x,k): gives the best approximation to the real x with denominator less or equal to kbezoutGGbezout(x,y): gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbezoutresbezoutres(x,y): gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbigomegabigomega(x): number of prime divisors of x, counted with multiplicitybinarybinary(x): gives the vector formed by the binary digits of x (x integer)binomialGLbinomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbitandbitand(x,y): bitwise "and" of two integers x and y. Negative numbers behave as if modulo big power of 2bitnegGD-1,L,bitneg(x,{n=-1}): bitwise negation of an integers x truncated to n bits. n=-1 means represent infinite sequences of bit 1 as negative numbers. Negative numbers behave as if modulo big power of 2bitnegimplybitnegimply(x,y): bitwise "negated imply" of two integers x and y, in other words, x BITAND BITNEG(y). Negative numbers behave as if modulo big power of 2bitorbitor(x,y): bitwise "or" of two integers x and y. Negative numbers behave as if modulo big power of 2bittestbittest(x,n): gives bit number n (coefficient of 2^n) of the integer x. Negative numbers behave as if modulo big power of 2bitxorbitxor(x,y): bitwise "exclusive or" of two integers x and y. Negative numbers behave as if modulo big power of 2bnfcertifylGbnfcertify(bnf): certify the correctness (i.e. remove the GRH) of the bnf data output by bnfclassunit or bnfinitbnfclassunitGD0,L,DGpbnfclassunit(P,{flag=0},{tech=[]}): compute the class group, regulator of the number field defined by the polynomial P, and also the fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units. See manual for details about tech. P may also be a non-zero integer, and is then considered as the discriminant of a quadratic orderbnfclgpGDGpbnfclgp(P,{tech=[]}): compute the class group of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfdecodemodulebnfdecodemodule(nf,fa): given a coded module fa as in bnrdisclist, gives the true modulebnfinitbnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units, 3: small bnfinit, which can be converted to a big one using bnfmake. See manual for details about techbnfisintnormbnfisintnorm(bnf,x): compute a complete system of solutions (modulo units of positive norm) of the absolute norm equation N(a)=x, where a belongs to the maximal order of big number field bnf (if bnf is not certified, this depends on GRH)bnfisnormGGD1,L,pbnfisnorm(bnf,x,{flag=1}): Tries to tell whether x (in Q) is the norm of some fractional y (in bnf). Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-unit, with S a certain list of primes (in bnf) containing (among others) all primes dividing x. If bnf is known to be Galois, set flag=0 (in this case, x is a norm iff b=1). If flag is non zero the program adds to S all the primes : dividing flag if flag<0, or less than flag if flag>0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois closure of bnfbnfisprincipalGGD1,L,bnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with flag<=2), gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. flag is optional, whose meaning is: 0: output only v; 1: default; 2: output only v, precision being doubled until the result is obtained; 3: as 2 but output generatorsbnfissunitGGGbnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu by bnfsunit, gives the column vector of exponents of x on the fundamental S-units and the roots of unity if x is a unit, the empty vector otherwisebnfisunitbnfisunit(bnf,x): bnf being output by bnfinit (with flag<=2), gives the column vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisebnfmakebnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfbnfnarrowbnfnarrow(bnf): given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class groupbnfregbnfreg(P,{tech=[]}): compute the regulator of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfsignunitbnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnfsunitbnfsunit(bnf,S): compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals. res[1] contains the S-units, res[5] the S-classgroup. See manual for detailsbnfunitbnfunit(bnf): compute the fundamental units of the number field bnf output by bnfinit when they have not yet been computed (i.e. with flag=2)bnrL1GDGD0,L,pbnrL1(bnr, {subgroup}, {flag=0}): bnr being output by bnrinit(,,1) and subgroup being a square matrix defining a congruence subgroup of bnr (the trivial subgroup if omitted), for each character of bnr trivial on this subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi) of the expansion at s = 0). The binary digits of flag mean 1: if 0 then compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in this case, only for non-trivial characters), 2: if 0 then compute the value of the primitive L-function associated to chi, if 1 then compute the value of the L-function L_S(s, chi) where S is the set of places dividing the modulus of bnr (and the infinite places), 3: return also the charactersbnrclassbnrclass(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this module. flag is optional, and can be 0: default, 1: compute data necessary for working in the ray class group, for example with functions such as bnrisprincipal or bnrdisc, without computing the generators of the ray class group, or 2: with the generators. When flag=1 or 2, the fifth component is the ray class group structure obtained when flag=0bnrclassnobnrclassno(bnf,x): ray class number of the module x for the big number field bnf. Faster than bnrclass if only the ray class number is wantedbnrclassnolistbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersbnrconductorGDGDGDGbnrconductor(a1,{a2},{a3},{flag=0}): conductor f of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc). flag is optional and can be 0: default, 1: returns [f, Cl_f, H], H subgroup of the ray class group modulo f defining the extension, 2: returns [f, bnr(f), H]bnrconductorofcharbnrconductorofchar(bnr,chi): conductor of the character chi on the ray class group bnrbnrdiscGDGDGD0,L,bnrdisc(a1,{a2},{a3},{flag=0}): absolute or relative [N,R1,discf] of the field defined by a1,a2,a3. [a1,{a2},{a3}] is of type [bnr], [bnr,subgroup], [bnf, module] or [bnf,module,subgroup], where bnf is as output by bnfclassunit (with flag<=2), bnr by bnrclass (with flag>0), and subgroup is the HNF matrix of a subgroup of the corresponding ray class group (if omitted, the trivial subgroup). flag is optional whose binary digits mean 1: give relative data; 2: return 0 if module is not the conductorbnrdisclistbnrdisclist(bnf,bound,{arch}): gives list of discriminants of ray class fields of all conductors up to norm bound, in a long vector The ramified Archimedean places are given by arch; all possible values are taken if arch is omitted. Supports the alternative syntax bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch (with units)bnrinitbnrinit(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data linked to the ray class group structure corresponding to this module. flag is optional, and can be 0: default (same as bnrclass with flag = 1), 1: compute also the generators (same as bnrclass with flag = 2). The fifth component is the ray class group structurebnrisconductorlGDGDGbnrisconductor(a1,{a2},{a3}): returns 1 if the modulus is the conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the only desired resultbnrisprincipalbnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. If (optional) flag is set to 0, output only vbnrrootnumberbnrrootnumber(bnr,chi,{flag=0}); returns the so-called Artin Root Number, i.e. the constant W appearing in the functional equation of the Hecke L-function associated to chi. Set flag = 1 if the character is known to be primitivebnrstarkbnrstark(bnr,{subgroup}): bnr being as output by bnrinit(,,1), finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the trivial subgroup if omitted) using Stark's units. The ground field and the class field must be totally real.breakD1,L,break({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopsceilceil(x): ceiling of x=smallest integer>=xcenterliftcenterlift(x,{v}): centered lift of x. Same as lift except for integermodschangevar(x,y): change variables of x according to the vector ycharpolyGDnD0,L,charpoly(A,{v=x},{flag=0}): det(v*Id-A)=characteristic polynomial of the matrix or polmod A. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the defaultchineseGDGchinese(x,{y}): x,y being both intmods (or polmods) computes z in the same residue classes as x and ycomponentcomponent(x,s): the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]. For list objects such as nf, bnf, bnr or ell, it is much easier to use member functions starting with "."concatconcat(x,{y}): concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to be lists). If y is omitted, x has to be a list or row vector and its elements are concatenatedconjconj(x): the algebraic conjugate of xconjvecconjvec(x): conjugate vector of the algebraic number xcontentcontent(x): gcd of all the components of x, when this makes sensecontfracGDGD0,L,contfrac(x,{b},{lmax}): continued fraction expansion of x (x rational,real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansioncontfracpnqncontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xcoreGD0,L,core(n,{flag=0}): unique (positive of negative) squarefree integer d dividing n such that n/d is a square. If (optional) flag is non-null, output the two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisccoredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)). If (optional) flag is non-null, output a two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercoscos(x): cosine of xcoshcosh(x): hyperbolic cosine of xcotancotan(x): cotangent of xdefaultD"",r,D"",s,D0,L,default({opt},{v}): returns the current value of the current default opt. If v is present, set opt to v first. If no argument is given, print a list of all defaults as well as their values.denominatordenominator(x): denominator of x (or lowest common denominator in case of an array)derivderiv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omitteddilogdilog(x): dilogarithm of xdirdivdirdiv(x,y): division of the Dirichlet series x by the Dirichlet series ydireulerV=GGEDGdireuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s). If c is present, output only the first c termsdirmuldirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series ydirzetakdirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1divisorsdivisors(x): gives a vector formed by the divisors of x in increasing orderdivremGGDndivrem(x,y,{v}): euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (to main variable if v is omitted)eint1eint1(x,{n}): exponential integral E1(x). If n is present, computes the vector of the first n values of the exponential integral E1(n.x) (x > 0)elladdelladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eellakellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellanellan(e,n): computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<2^24 on a 32-bit machine)ellapellap(e,p,{flag=0}): computes a_p for the elliptic curve e using Shanks-Mestre's method. flag is optional and can be set to 0 (default) or 1 (use Jacobi symbols)ellbilGGGpellbil(e,z1,z2): canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsellchangecurveellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]ellchangepointellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]ellconvertnameellconvertname(name): convert an elliptic curve name (as found in the elldata database) from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name.elleisnumelleisnum(om,k,{flag=0}): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalizationelletaelleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellgeneratorsellgenerators(E): if E is an elliptic curve as output by ellinit(), return the generators of the Mordell-Weil group associated to the curve. This function depends on the curve being referenced in the elldata database.ellglobalredellglobalred(e): e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pellheightGGD2,L,pellheight(e,x,{flag=2}): canonical height of point x on elliptic curve E defined by the vector e. flag is optional and selects the algorithm used to compute the archimedean local height. Its meaning is 0: use theta-functions, 1: use Tate's method, 2: use Mestre's AGMellheightmatrixellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionsellidentifyellidentify(E): look up the elliptic curve E in the elldata database and return [[N, M, ...], C] where N is the name of the curve in J. E. Cremona database, M the minimal model and C the coordinates change (see ellchangecurve).ellinitGD0,L,pellinit(x,{flag=0}): x being the vector [a1,a2,a3,a4,a6] defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.X + a6, gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,disc,j,[e1,e2,e3],w1,w2,eta1,eta2,area]. If the curve is defined over a p-adic field, the last six components are replaced by root,u^2,u,q,w,0. If optional flag is 1, omit them altogether. x can also be a string, in this case the coefficients of the curve with matching name are looked in the elldata database if available.ellisoncurveellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notelljellj(x): elliptic j invariant of xelllocalredelllocalred(e,p): e being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the Kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pelllseriesGGDGpelllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1ellminimalmodelGD&ellminimalmodel(e,{&v}): return the standard minimal integral model of the rational elliptic curve e. Sets v to the corresponding change of variablesellorderellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellordinateellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellpointtozellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellpowellpow(e,x,n): n times the point x on elliptic curve e (n in Z)ellrootnolGDGellrootno(e,{p=1}): root number for the L-function of the elliptic curve e. p can be 1 (default), global root number, or a prime p (including 0) for the local root number at pellsearchellsearch(N): if N is an integer, it is taken as a conductor else if N is a string, it can be a curve name ("11a1"), a isogeny class ("11a") or a conductor ("11"). Return all curves in the elldata database that match the  property.ellsigmaellsigma(om,z,{flag=0}): om=[om1,om2], value of the Weierstrass sigma function of the lattice generated by om at z if flag = 0 (default). If flag = 1, arbitrary determination of the logarithm of sigma. If flag = 2 or 3, same but using the product expansion instead of theta seriesellsubellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eelltaniyamaGPelltaniyama(e): modular parametrization of elliptic curve eelltorselltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-NagellellwpGDGD0,L,pPellwp(e,{z=x},{flag=0}): Complex value of Weierstrass P function at z on the lattice generated over Z by e=[om1,om2] (e as given by ellinit is also accepted). Optional flag means 0 (default), compute only P(z), 1 compute [P(z),P'(z)], 2 consider om as an elliptic curve and compute P(z) for that curve (identical to ellztopoint in that case). If z is omitted or is a simple variable, return formal expansion in zellzetaellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zellztopointellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zerfcerfc(x): complementary error functionerrorvs*error("msg"): abort script with error message msgetaeta(x,{flag=0}): if flag=0, eta function without the q^(1/24), otherwise eta of the complex number x in the upper half plane intelligently computed using SL(2,Z) transformationseulerphieulerphi(x): Euler's totient function of xevaleval(x): evaluation of x, replacing variables by their valueexpexp(x): exponential of xfactorfactor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)factorbackGDGDGfactorback(f,{e},{nf}): given a factorisation f, gives the factored object back. If this is a prime ideal factorisation you must supply the corresponding number field as last argument. If e is present, f has to be a vector of the same length, and we return the product of the f[i]^e[i]factorcantorfactorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausfactorfffactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factorialfactorial(x): factorial of x (x C-integer), the result being given as a real numberfactorintfactorint(x,{flag=0}): factor the integer x. flag is optional, whose binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM (huge composites will be declared prime)factormodfactormod(x,p,{flag=0}): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are givenfactornffactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tfactorpadicGGLD0,L,factorpadic(x,p,r,{flag=0}): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)ffinitGLDnffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]fibonaccifibonacci(x): fibonacci number of index x (x C-integer)floorfloor(x): floor of x = largest integer<=xforvV=GGIfor(X=a,b,seq): the sequence is evaluated, X going from a up to bfordivvGVIfordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nforellvVLLIforell(E,a,b,seq): execute seq for each elliptic curves E of conductor between a and b in the elldata database.forprimeforprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bforstepvV=GGGIforstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b in steps of s (can be a vector of steps)forsubgroupvV=GDGIforsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the abelian group G (in SNF form), whose index is bounded by bound. H is given as a left divisor of G in HNF formforvecvV=GID0,L,forvec(x=v,seq,{flag=0}): v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2), restrict to increasing (resp. strictly increasing) sequencesfracfrac(x): fractional part of x = x-floor(x)galoisexportgaloisexport(gal,{flag}): gal being a galois field as output by galoisinit, output a string representing the underlying permutation group in GAP notation (default) or Magma notation (flag = 1)galoisfixedfieldGGD0,L,Dngaloisfixedfield(gal,perm,{flag},{v=y}): gal being a galois field as output by galoisinit and perm an element of gal.group or a vector of such elements, return [P,x] such that P is a polynomial defining the fixed field of gal[1] by the subgroup generated by perm, and x is a root of P in gal expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 return [P,x,F] where F is the factorization of gal.pol over the field defined by P, where the variable v stands for a root of Pgaloisidentifygaloisidentify(gal): gal being a galois field as output by galoisinit, output the isomorphism class of the underlying abstract group as a two-components vector [o,i], where o is the group order, and i is the group index in the GAP4 small group librarygaloisinitgaloisinit(pol,{den}): pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group and all neccessary informations for computing fixed fields. den is optional and has the same meaning as in nfgaloisconj(,4)(see manual)galoisisabeliangaloisisabelian(gal,{flag=0}): gal being as output by galoisinit, return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if flag=0, 1 if flag is 1, and the SNF of gal is flag=2galoispermtopolgaloispermtopol(gal,perm): gal being a galois field as output by galoisinit and perm a element of gal.group, return the polynomial defining the corresponding Galois automorphismgaloissubcycloGDGD0,L,Dngaloissubcyclo(N,H,{fl=0},{v}):Compute a polynomial (in variable v) defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)*. N can be an integer n, znstar(n) or bnrinit(bnfinit(y),[n,[1]],1). H can be given by a generator, a set of generator given by a vector or a HNF matrix (see manual). If flag is 1, output only the conductor of the abelian extension. If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.galoissubfieldsGD0,L,Dngaloissubfields(G,{flags=0},{v}):Output all the subfields of G. flags have the same meaning as for galoisfixedfieldgaloissubgroupsgaloissubgroups(G):Output all the subgroups of Ggammagamma(x): gamma function at xgammahgammah(x): gamma of x+1/2 (x integer)gcd(x,{y}): greatest common divisor of x and y.getheapgetheap(): 2-component vector giving the current number of objects in the heap and the space they occupygetrandlgetrand(): current value of random number seedgetstackgetstack(): current value of stack pointer avmagettimegettime(): time (in milliseconds) since last call to gettimeglobalglobal(x): declare x to be a global variablehilbertlGGDGhilbert(x,y,{p}): Hilbert symbol at p of x,y. If x,y are integermods or p-adic, p can be omittedhyperuhyperu(a,b,x): U-confluent hypergeometric functionidealaddidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealaddtooneidealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1. Otherwise, x and y are ideals, and if they sum up to 1, find one element in each of them such that the sum is 1idealappridealappr(nf,x,{flag=0}): x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other p. If (optional) flag is non-null x must be a prime ideal factorization with possibly zero exponentsidealchineseidealchinese(nf,x,y): x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprimeidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yidealdivGGGD0,L,idealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF in the number field nf. If (optional) flag is non-null, the quotient is supposed to be an integral ideal (slightly faster)idealfactoridealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfidealhnfidealhnf(nf,a,{b}): hermite normal form of the ideal a in the number field nf, whatever form a may have. If called as idealhnf(nf,a,b), the ideal is given as aZ_K+bZ_K in the number field K defined by nfidealintersectidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealinvidealinv(nf,x,{flag=0}): inverse of the ideal x in the number field nf. If flag is omitted or set to 0, use the different. If flag is 1 do not use itideallistGLD4,L,ideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of all ideals of norm<=bound. If (optional) flag is present, its binary digits are toggles meaning 1: give generators; 2: add units; 4: give only the ideals and not the bid.ideallistarchideallistarch(nf,list,arch): list is a vector of vectors of of bid's as output by ideallist. Return a vector of vectors with the same number of components as the original list. The leaves give information about moduli whose finite part is as in original list, in the same order, and archimedean part is now arch. The information contained is of the same kind as was present in the input.ideallogideallog(nf,x,bid): if bid is a big ideal, as given by idealstar(nf,I,1) or idealstar(nf,I,2), gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)idealminidealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfidealmulGGGD0,L,pidealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the number field nf. If (optional) flag is non-nul, reduce the resultidealnormidealnorm(nf,x): norm of the ideal x in the number field nfidealpowidealpow(nf,x,n,{flag=0}): n-th power of the ideal x in HNF in the number field nf If (optional) flag is non-null, reduce the resultidealprimedecidealprimedec(nf,p): prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a. Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantidealprincipalidealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfidealredidealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealstaridealstar(nf,I,{flag=1}): gives the structure of (Z_K/I)^*. flag is optional, and can be 0: simply gives the structure as a 3-component vector v such that v[1] is the order (i.e. eulerphi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators. If flag=1 (default), gives idealstarinit, i.e. a 6-component vector [I,v,fa,f2,U,V] where v is as above without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*. Finally if flag=2, same as with flag=1 except that the generators are also givenidealtwoeltidealtwoelt(nf,x,{a}): two-element representation of an ideal x in the number field nf. If (optional) a is non-zero, first element will be equal to aidealvallGGGidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfideleprincipalideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfifif(a,seq1,seq2): if a is nonzero, seq1 is evaluated, otherwise seq2. seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can be omitted alsoimagimag(x): imaginary part of xincgamincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)incgamcincgamc(s,x): complementary incomplete gamma functionintcircV=GGEDGpintcirc(X=a,R,s,{tab}): numerical integration of s on the circle  |z-a|=R, divided by 2*I*Pi. tab is as in intnum.intformalintformal(x,{y}): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omittedintfouriercosV=GGGEDGpintfouriercos(X=a,b,x,s,{tab}): numerical integration from a to b of cos(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the cosine-Fourier transform if a=-infty and b=+infty.intfourierexpintfourierexp(X=a,b,x,s,{tab}): numerical integration from a to b of exp(-2*I*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the ordinary Fourier transform if a=-infty and b=+infty. Note the minus sign.intfouriersinintfouriersin(X=a,b,x,s,{tab}): numerical integration from a to b of sin(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the sine-Fourier transform if a=-infty and b=+infty.intfuncinitV=GGED0,L,D0,L,pintfuncinit(X=a,b,s,{flag=0},{m=0}): initialize tables for integrations  from a to b using a weight s(X). Essential for integral transforms such as intmellininv, intlaplaceinv and intfourier, since it avoids recomputing all the time the same quantities. Must then be used with intmellininvshort (for intmellininv) and directly with intnum and not with the corresponding  integral transforms for the others. See help for intnum for coding of a  and b, and m is as in intnuminit. If flag is nonzero, assumes that  s(-X)=conj(s(X)), which is twice faster.intlaplaceinvintlaplaceinv(X=sig,x,s,{tab}): numerical integration on the line real(z) = sig of s(z)exp(xz)dz/(2*I*Pi), i.e. inverse Laplace transform of s at x. tab is as in intnum.intmellininvintmellininv(X=sig,x,s,{tab}): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin  transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). tab is as in intnum. Use  intmellininvshort if several values must be computed.intmellininvshortintmellininvshort(sig,x,tab): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). Compulsory table tab has been  precomputed using the command intfuncinit(t=[[-1],sig[2]],[[1],sig[2]],s)  (with possibly its two optional additional parameters), where sig[2] = 1 if not given. Orders of magnitude faster than intmellininv.intnumintnum(X=a,b,s,{tab}): numerical integration of s from a to b with  respect to X. a (and similarly b) is coded as follows. It can be a scalar: f is assumed to be C^infty at a. It can be a two component vector [a[1],a[2]], where a[1] is the scalar, and a[2] is the singularity exponent (in ]-1,0]), logs being neglected. It can be a one component vector [1] or [-1] meaning +infty or -infty, slowly decreasing functions. It can be a two component vector [[1], z] or [[-1], z], where [1] or [-1] indicates +infty or -infty and z is coded as follows. If z is zero, slowly decreasing. If z is real positive, exponentially decreasing, of the type exp(-zX). If z<-1, very slowly decreasing like X^(-z). If z is complex nonreal, real part is ignored and if z = r+I*s then if s>0, cosine oscillation exactly cos(sX), while if s<0, sine oscillation exactly sin(sX). If f is exponentially decreasing times oscillating function, you have a choice, but it is in general better to choose the oscillating part. Finally tab is either 0 (let the program choose  the integration step), a positive integer m (choose integration step 1/2^m), or a table tab precomputed with intnuminit (depending on the type of interval: compact, semi-compact or R, very slow, slow, exponential, or cosine or sine-oscillating decrease).intnuminitintnuminit(a,b,{m=0}): initialize tables for integrations from a to b. See help for intnum for coding of a and b. Possible types: compact interval, semi-compact (one extremity at + or - infinity) or R, and very slowly, slowly or exponentially decreasing, or sine or cosine oscillating at infinities,intnuminitgenVGGED0,L,D0,L,pintnuminitgen(t,a,b,ph,{m=0},{flag=0}): initialize tables for  integrations from a to b using abcissas ph(t) and weights ph'(t). Note that  there is no equal sign after the variable name t since t always goes from  -infty to +infty, but it is ph(t) which goes from a to b, and this is not  checked. If flag = 1 or 2, multiply the reserved table length by 4^flag, to  avoid corresponding error.intnumrombV=GGED0,L,pintnumromb(X=a,b,s,{flag=0}): numerical integration of s (smooth in  ]a,b[) from a to b with respect to X. flag is optional and mean 0: default.  s can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be  plus or minus infinity (chosen suitably), but of same sign; 3: s has only  limits at a or bintnumsteplpintnumstep(): gives the default value of m used by all intnum and sumnum  routines, such that the integration step is 1/2^m.isfundamentalisfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notispowerlGDGD&ispower(x,{k},{&n}): true (1) if x is a k-th power, false (0) if not. If n is given and a k-th root was computed in the process, put that in n. If k is omitted, return the maximal k >= 2 such that x = n^k is a perfect power, or 0 if no such k exist.isprimeisprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of algorithms. If flag is 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2, the primality is certified using the APRCL test.ispseudoprimeispseudoprime(x,{n}): true(1) if x is a strong pseudoprime, false(0) if not. If n is 0 or omitted, use BPSW test, otherwise use strong Rabin-Miller test for n randomly chosen basesissquareissquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is given puts the exact square root there if it was computedissquarefreeissquarefree(x): true(1) if x is squarefree, false(0) if notkillvSkill(x): kills the present value of the variable or function x. Returns new value or 0kroneckerkronecker(x,y): kronecker symbol (x/y)lcmlcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)lengthlength(x): number of non code words in x, number of characters for a stringlexiGGlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)liftlift(x,{v}): lifts every element of Z/nZ to Z or T[x]/PT[x] to T[x] for a type T if v is omitted, otherwise lift only polymods with main variable v. If v does not occur in x, lift only intmodslindeplindep(x,{flag=0}): Z-linear dependencies between components of x. flag is optional, and can be 0: default, PSLQ; -1: using Hastad et al; -2: returns a non-trivial kernel vector (not integral in general); positive, and in that case should be between 0.5 and 1.0 times the accuracy in decimal digits of x, using a standard LLLlistcreatelistcreate(n): creates an empty list of maximum length nlistinsertGGLlistinsert(list,x,n): insert x at index n in list, shifting the remaining elements to the rightlistkillvGlistkill(list): kills listlistputlistput(list,x,{n}): sets n-th element of list equal to x. If n is omitted or greater than the current list length, just append xlistsortlistsort(list,{flag=0}): sort list in place. If flag is non-zero, suppress all but one occurence of each element in listlngammalngamma(x): logarithm of the gamma function of xloglog(x): natural logarithm of x.matadjointmatadjoint(x): adjoint matrix of xmatalgtobasismatalgtobasis(nf,x): nfalgtobasis applied to every element of the matrix xmatbasistoalgmatbasistoalg(nf,x): nfbasistoalg applied to every element of the matrix xmatcompanionmatcompanion(x): companion matrix to polynomial xmatdetmatdet(x,{flag=0}): determinant of the matrix x using Gauss-Bareiss. If (optional) flag is set to 1, use classical gaussian elimination (slightly better for integer entries)matdetintmatdetint(x): some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with mathnfmodmatdiagonalmatdiagonal(x): creates the diagonal matrix whose diagonal entries are the entries of the vector xmateigenmateigen(x): eigenvectors of the matrix x given as columns of a matrixmatfrobeniusmatfrobenius(M,{flag},{v=x}): Return the Frobenius form of the square matrix M. If flag is 1, return only the elementary divisors as a vector of polynomials in the variable v. If flag is 2, return a two-components vector [F,B] where F is the Frobenius form and B is the basis change so that M=B^-1*F*B.mathessmathess(x): Hessenberg form of xmathilbertmathilbert(n): Hilbert matrix of order n (n C-integer)mathnfmathnf(A,{flag=0}): (upper triangular) Hermite normal form of A, basis for the lattice formed by the columns of A. flag is optional whose value range from 0 to 4 (0 if omitted), meaning : 0: naive algorithm. 1: Use Batut's algorithm. Output 2-component vector [H,U] such that H is the HNF of A, and U is a unimodular matrix such that AU=H. 3: Use Batut's algorithm. Output [H,U,P] where P is a permutation matrix such that P A U = H. 4: as 1, using a heuristic variant of LLL reduction along the waymathnfmodmathnfmod(x,d): (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is a multiple of the non-zero determinant of this latticemathnfmodidmathnfmodid(x,d): (upper triangular) Hermite normal form of x concatenated with d times the identity matrixmatidmatid(n): identity matrix of order n (n C-integer)matimagematimage(x,{flag=0}): basis of the image of the matrix x. flag is optional and can be set to 0 or 1, corresponding to two different algorithmsmatimagecomplmatimagecompl(x): vector of column indices not corresponding to the indices given by the function matimagematindexrankmatindexrank(x): gives two extraction vectors (rows and columns) for the matrix x such that the extracted matrix is square of maximal rankmatintersectmatintersect(x,y): intersection of the vector spaces whose bases are the columns of x and ymatinverseimagematinverseimage(x,y): an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwisematisdiagonalmatisdiagonal(x): true(1) if x is a diagonal matrix, false(0) otherwisematkermatker(x,{flag=0}): basis of the kernel of the matrix x. flag is optional, and may be set to 0: default; non-zero: x is known to have integral entriesmatkerintmatkerint(x,{flag=0}): LLL-reduced Z-basis of the kernel of the matrix x with integral entries. flag is optional, and may be set to 0: default, uses a modified LLL, 1: uses matrixqzmatmuldiagonalmatmuldiagonal(x,d): product of matrix x by diagonal matrix whose diagonal coefficients are those of the vector d, equivalent but faster than x*matdiagonal(d)matmultodiagonalmatmultodiagonal(x,y): product of matrices x and y, knowing that the result will be a diagonal matrix. Much faster than general multiplication in that casematpascalLDGmatpascal(n,{q}): Pascal triangle of order n if q is omited. q-Pascal triangle otherwisematrankmatrank(x): rank of the matrix xmatrixGGDVDVDImatrix(m,n,{X},{Y},{expr=0}): mXn matrix of expression expr, the row variable X going from 1 to m and the column variable Y going from 1 to n. By default, fill with 0smatrixqzmatrixqz(x,p): if p>=0, transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwise. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned by the columns of x. If p=-2, finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsizematsize(x): number of rows and columns of the vector/matrix x as a 2-vectormatsnfmatsnf(x,{flag=0}): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in dmatsolvematsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)matsolvemodmatsolvemod(M,D,B,{flag=0}): one solution of system of congruences MX=B mod D (M matrix, B and D column vectors). If (optional) flag is non-null return all solutionsmatsupplementmatsupplement(x): supplement the columns of the matrix x to an invertible matrixmattransposemattranspose(x): x~=transpose of xmaxmax(x,y): maximum of x and yminmin(x,y): minimum of x and yminpolyminpoly(A,{v=x}): minimal polynomial of the matrix or polmod A.modreversemodreverse(x): reverse polymod of the polymod x, if it existsmoebiusmoebius(x): Moebius function of xnewtonpolynewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime pnextnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopsnextprimenextprime(x): smallest pseudoprime >= xnfalgtobasisnfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zknfbasisGD0,L,DGnfbasis(x,{flag=0},{p}): integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithm. Second and third args are optional. Binary digits of flag mean 1: assume that no square of a prime>primelimit divides the discriminant of x, 2: use round 2 algorithm instead. If present, p provides the matrix of a partial factorization of the discriminant of x, useful if one wants only an order maximal at certain primes onlynfbasistoalgnfbasistoalg(nf,x): transforms the column vector x on the integral basis into an algebraic numbernfdetintnfdetint(nf,x): multiple of the ideal determinant of the pseudo generating set xnfdiscnfdisc(x,{flag=0},{p}): discriminant of the number field defined by the polynomial x using round 4. Optional args flag and p are as in nfbasisnfeltdivnfeltdiv(nf,a,b): element a/b in nfnfeltdiveucnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltdivmodprGGGGnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltdivremnfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfeltmodnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernfeltmulnfeltmul(nf,a,b): element a. b in nfnfeltmulmodprnfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltpownfeltpow(nf,a,k): element a^k in nfnfeltpowmodprnfeltpowmodpr(nf,a,k,pr): element a^k modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltreducenfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r is smallnfeltreducemodprnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltvalnfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnffactornffactor(nf,x): factor polynomial x in number field nfnffactormodnffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr in number field nfnfgaloisapplynfgaloisapply(nf,aut,x): Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfnfgaloisconjnfgaloisconj(nf,{flag=0},{den}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 2 : use complex numbers, LLL on integral basis (not always complete); 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for detail). nf can be simply a polynomial with flag 0,2 and 4, meaning: 0: use combination of flag 4 and 2, not always complete (but a warning is issued when the list is not proven complete); 2 & 4: same meaning and restrictions. Note that only flag 4 can be applied to fields of large degrees (approx. >= 20)nfhilbertlGGGDGnfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1 otherwise. Otherwise compute the local symbol modulo the prime ideal pnfhnfnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhnfmodnfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfinitnfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial, gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]. flag is optional and can be set to 0: default; 1: do not compute different; 2: first use polred to find a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2 and Mod(a,P) is a polymod equal to Mod(x,pol) and P=nf.pol; 4: as 2 but use a partial polred; 5: is to 3 what 4 is to 2nfisideallGGnfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notnfisinclnfisincl(x,y): tests whether the number field x is isomorphic to a subfield of y (where x and y are either polynomials or number fields as output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y is a number field, a faster algorithm is usednfisisomnfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfkermodprnfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in modpr format (see nfmodprinit)nfmodprinitnfmodprinit(nf,pr): transform the 5 element row vector pr representing a prime ideal into modpr format necessary for all operations mod pr in the number field nf (see manual for details about the format)nfnewprecnfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionnfrootsDGGnfroots({nf},pol): roots of polynomial pol belonging to nf (Q if omitted) without multiplicitynfrootsof1nfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfnfsnfnfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfsolvemodprnfsolvemodpr(nf,a,b,pr): solution of a*x=b in Z_K/pr, where a is a matrix and b a column vector, and where pr is in modpr format (see nfmodprinit)nfsubfieldsnfsubfields(nf,{d=0}): find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nfnormnorm(x): norm of xnorml2norml2(x): square of the L2-norm of the vector xnumbpartnumbpart(x): number of partitions of xnumdivnumdiv(x): number of divisors of xnumeratornumerator(x): numerator of xnumtopermLGnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)omegaomega(x): number of distinct prime divisors of xpadicapprpadicappr(x,a): p-adic roots of the polynomial x congruent to a mod ppadicprecpadicprec(x,p): absolute p-adic precision of object xpermtonumpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolcoeffpolcoeff(x,s,{v}): coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[]). With respect to the main variable if v is omitted, with respect to the variable v otherwisepolcompositumpolcompositum(pol1,pol2,{flag=0}): vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2. If (optional) flag is set (i.e non-null), output for each compositum, not only the compositum polynomial pol, but a vector [pol,al1,al2,k] where al1 (resp. al2) is a root of pol1 (resp. pol2) expressed as a polynomial modulo pol, and a small integer k such that al2+k*al1 is the chosen root of polpolcycloLDnpolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)poldegreelGDnpoldegree(x,{v}): degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwise. For scalar x, return 0 is x is non-zero and a negative number otherwisepoldiscpoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepoldiscreducedpoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolgaloispolgalois(x): Galois group of the polynomial x (see manual for group coding). Return [n, s, k, name] where n is the order, s the signature, k the index and name is the GAP4 name of the transitive group.polhenselliftGGGLpolhensellift(x, y, p, e): lift the factorization y of x modulo p to a factorization modulo p^e using Hensel lift. The factors in y must be pairwise relatively prime modulo ppolinterpolateGDGDGD&polinterpolate(xa,{ya},{x},{&e}): polynomial interpolation at x according to data vectors xa, ya (ie return P such that P(xa[i]) = ya[i] for all i). If ya is omitter, return P such that P(i) = xa[i]. If present, e will contain an error estimate on the returned valuepolisirreduciblepolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantpolleadpollead(x,{v}): leading coefficient of polynomial or series x, or x itself if x is a scalar. Error otherwise. With respect to the main variable of x if v is omitted, with respect to the variable v otherwisepollegendrepollegendre(n,{v=x}): legendre polynomial of degree n (n C-integer), in variable vpolrecippolrecip(x): reciprocal polynomial of xpolredpolred(x,{flag=0},{p}): reduction of the polynomial x (gives minimal polynomials only). Second and third args are optional. The following binary digits of flag are significant 1: partial reduction, 2: gives also elements. p, if present, contains the complete factorization matrix of the discriminantpolredabspolredabs(x,{flag=0}): a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 norm. flag is optional, whose binary digit mean 1: give the element whose characteristic polynomial is the given polynomial. 4: give all polynomials of minimal T2 norm (give only one of P(x) and P(-x)). 16: partial reductionpolredordpolredord(x): reduction of the polynomial x, staying in the same orderpolresultantGGDnD0,L,polresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the variable v otherwise. flag is optional, and can be 0: default, assumes that the polynomials have exact entries (uses the subresultant algorithm), 1 for arbitrary polynomials, using Sylvester's matrix, or 2: using a Ducos's modified subresultant algorithmpolrootspolroots(x,{flag=0}): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage's method modified by Gourdon, or 1: uses a modified Newton methodpolrootsmodpolrootsmod(x,p,{flag=0}): roots mod p of the polynomial x. flag is optional, and can be 0: default, or 1: use a naive search, useful for small ppolrootspadicpolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsturmpolsturm(x,{a},{b}): number of real roots of the polynomial x in the interval]a,b] (which are respectively taken to be -oo or +oo when omitted)polsubcycloLLDnpolsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n). Output is a polynomial or a vector of polynomials is there are several such fields, or none.polsylvestermatrixpolsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowspolsympolsym(x,n): vector of symmetric powers of the roots of x up to npoltchebipoltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vpoltschirnhauspoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xpolylogLGD0,L,ppolylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of xpolzagierLLpolzagier(n,m): Zagier's polynomials of index n,mprecisionprecision(x,{n}): change the precision of x to be n (n C-integer). If n is omitted, output real precision of object xprecprimeprecprime(x): largest pseudoprime <= x, 0 if x<=1primeprime(n): returns the n-th prime (n C-integer)primepiprimepi(x): the prime counting function pi(x) = #{p <= x, p prime}.primesprimes(n): returns the vector of the first n primes (n C-integer)printprint(a): outputs a (in raw format) ending with newlineprint1print1(a): outputs a (in raw format) without ending with newlineprintpprintp(a): outputs a (in beautified format) ending with newlineprintp1printp1(a): outputs a (in beautified format) without ending with newlineprinttexprinttex(a): outputs a in TeX formatprodprod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionprodeulerV=GGEpprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionprodinfV=GED0,L,pprodinf(X=a,expr,{flag=0}): infinite product (X goes from a to infinity) of real or complex expression. flag can be 0 (default) or 1, in which case compute the product of the 1+expr insteadpsipsi(x): psi-function at xqfbclassnoqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsqfbcomprawqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yqfbhclassnoqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfbnucompqfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupowqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplqfbpowrawqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xqfbprimeformqfbprimeform(x,p): returns the prime form of discriminant x, whose first coefficient is pqfbredGD0,L,DGDGDGqfbred(x,{flag=0},{D},{isqrtD},{sqrtD}): reduction of the binary quadratic form x. All other args. are optional. D, isqrtD and sqrtD, if present, supply the values of the discriminant, floor(sqrt(D)) and sqrt(D) respectively. If D<0, its value is not used and all references to Shanks's distance hereafter are meaningless. flag can be any of 0: default, uses Shanks's distance function d; 1: use d, do a single reduction step; 2: do not use d; 3: do not use d, single reduction step.qfbsolveqfbsolve(Q,p): Return [x,y] so that Q(x,y)=p where Q is a binary quadratic form and p a prime number, or 0 if there is no solution.qfgaussredqfgaussred(x): square reduction of the (symmetric) matrix x (returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)qfjacobiqfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqflllqflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). The columns of x must be linearly independent, unless specified otherwise below. flag is optional,  and can be 0: default, 1: assumes x is integral, columns may be dependent, 2: assumes x is integral, returns a partially reduced basis, 4: assumes x is  integral, returns [K,I] where K is the integer kernel of x and I the LLL reduced image, 5: same as 4 but x may have polynomial coefficients, 8: same as 0 but x may have polynomial coefficientsqflllgramqflllgram(x,{flag=0}): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 8: lllgramgen, same as qflllgram when the coefficients are polynomialsqfminimGDGDGD0,L,pqfminim(x,{bound},{maxnum},{flag=0}): number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but uses a more robust, slower implementation, valid for non integral quadratic formsqfperfectionqfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aqfrepqfrep(x,B,{flag=0}): vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form x. Binary digits of flag mean 1: count vectors of even norm from 1 to 2B, 2: return a t_VECSMALL instead of a t_VECqfsignqfsign(x): signature of the symmetric matrix xquadclassunitquadclassunit(D,{flag=0},{tech=[]}): compute the structure of the class group and the regulator of the quadratic field of discriminant D. If flag is non-null (and D>0), compute the narrow class group. See manual for the optional technical parametersquaddiscquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadgenquadgen(x): standard generator of quadratic order of discriminant xquadhilbertquadhilbert(D,{pq}): relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf). If D<0, pq (if supplied) is a 2-component vector [p,q], where p,q are the prime numbers needed for Schertz's method. In that case, return 0 if [p,q] not suitable.quadpolyquadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadrayquadray(D,f,{lambda}): relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a bnf). For D < 0, lambda (if supplied) is the technical element of bnf  necessary for Schertz's method. In that case, return 0 if lambda is not suitable.quadregulatorquadregulator(x): regulator of the real quadratic field of discriminant xquadunitquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positiverandomrandom({N=2^31}): random integer between 0 and N-1readvecD"",s,readvec({filename}): create a vector whose components are the evaluation of all the expressions found in the input file filenamerealreal(x): real part of xremoveprimesremoveprimes({x=[]}): remove primes in the vector x (with at most 100 components) from the prime table. x can also be a single integer. List the current extra primes if x is omittedreorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesreturnreturn({x=0}): return from current subroutine with result xrnfalgtobasisrnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a relative numberfieldrnfbasisrnfbasis(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfbasistoalgrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldrnfcharpolyGGGDnrnfcharpoly(nf,T,alpha,{var=x}): characteristic polynomial of alpha over nf, where alpha belongs to the algebra defined by T over nf. Returns a polynomial in variable var (x by default)rnfconductorrnfconductor(bnf,polrel,{flag=0}): conductor of the Abelian extension of bnf defined by polrel. The result is [conductor,rayclassgroup,subgroup], where conductor is the conductor itself, rayclassgroup the structure of the corresponding full ray class group, and subgroup the HNF defining the norm group (Artin or Takagi group) on the given generators rayclassgroup[3]. If flag is non-zero, check that polrel indeed defines an Abelian extensionrnfdedekindrnfdedekind(nf,T,pr): relative Dedekind criterion over nf, applied to the order defined by a root of irreducible polynomial T, modulo the prime ideal pr. Returns [flag,basis,val], where basis is a pseudo-basis of the enlarged order, flag is 1 iff this order is pr-maximal, and val is the valuation in pr of the order discriminantrnfdetrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfdiscrnfdisc(nf,pol): given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfeltabstorelrnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfeltdownrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisernfeltreltoabsrnfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfeltuprnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldrnfequationrnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf, gives the absolute equation apol of the number field defined by pol. flag is optional, and can be 0: default, or non-zero, gives [apol,th], where th expresses the root of nf.pol in terms of the root of apolrnfhnfbasisrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfidealabstorelrnfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfidealdownrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfidealhnfrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldrnfidealmulrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldrnfidealnormabsrnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealnormrelrnfidealnormrel(rnf,x): relative norm of the ideal xrnfidealreltoabsrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationrnfidealtwoeltrnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf is a relative numberfieldrnfidealuprnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfinitrnfinit(nf,pol): pol being a non constant irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working in relative number fields (rnf functions). See manual for technical detailsrnfisfreernfisfree(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnfisnormrnfisnorm(T,x,{flag=0}): T is as output by rnfisnorminit applied to L/K. Tries to tell whether x is a norm from L/K. Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-integer, with S a list of places in K containing the ramified primes, generators of the class group of ext, as well as those primes dividing x. If L/K is Galois, omit flag, otherwise it is used to add more places to S: all the places above the primes p <= flag (resp. p | flag) if flag > 0 (resp. flag < 0). The answer is guaranteed (i.e x norm iff b=1) if L/K is Galois or, under GRH, if S contains all primes less than 12.log(disc(M))^2, where M is the normal closure of L/KrnfisnorminitGGD2,L,rnfisnorminit(pol,polrel,{flag=2}): let K be defined by a root of pol, L/K the extension defined by polrel. Compute technical data needed by rnfisnorm to solve norm equations Nx = a, for x in L, and a in K. If flag=0, do not care whether L/K is Galois or not; if flag = 1, assume L/K is Galois; if flag = 2, determine whether L/K is Galoisrnfkummerrnfkummer(bnr,{subgroup},{deg=0}): bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the ray class field if subgroup is omitted). deg can be zero (default), or positive, and in this case the output is the list of all relative equations of degree deg for the given bnrrnflllgramrnflllgram(nf,pol,order): given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfnormgrouprnfnormgroup(bnr,polrel): norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by polrel, where the module corresponding to bnr is assumed to be a multiple of the conductor. The result is the HNF defining the norm group on the given generators in bnr[5][3]rnfpolredrnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplerrnfpolredabsrnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf, finds a relative simpler polynomial defining the same field. Binary digits of flag mean: 1: return also the element whose characteristic polynomial is the given polynomial, 2: return an absolute polynomial, 16: partial reductionrnfpseudobasisrnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitzrnfsteinitz(nf,order): given an order as output by rnfpseudobasis, gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialroundround(x,{&e}): take the nearest integer to all the coefficients of x. If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitsserconvolserconvol(x,y): convolution (or Hadamard product) of two power seriesserlaplaceserlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n. For the reverse operation, use serconvol(x,exp(X))serreverseserreverse(x): reversion of the power series xsetintersectsetintersect(x,y): intersection of the sets x and ysetissetsetisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsetminussetminus(x,y): set of elements of x not belonging to ysetrandlLsetrand(n): reset the seed of the random number generator to nsetsearchlGGD0,L,setsearch(x,y,{flag=0}): looks if y belongs to the set x. If flag is 0 or omitted, returns 0 if it is not, otherwise returns the index j such that y==x[j]. If flag is non-zero, return 0 if y belongs to x, otherwise the index j where it should be insertedsetunionsetunion(x,y): union of the sets x and yshiftshift(x,n): shift x left n bits if n>=0, right -n bits if n<0.shiftmulshiftmul(x,n): multiply x by 2^n (n>=0 or n<0)sigmaGD1,L,sigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is optional and if omitted is assumed to be equal to 1signiGsign(x): sign of x, of type integer, real or fractionsimplifysimplify(x): simplify the object x as much as possiblesinsin(x): sine of xsinhsinh(x): hyperbolic sine of xsizebytesizebyte(x): number of bytes occupied by the complete tree of the object xsizedigitsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsolvesolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sqrsqr(x): square of x. NOT identical to x*xsqrtsqrt(x): square root of xsqrtintsqrtint(x): integer square root of x (x integer)sqrtnGGD&psqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is set to a suitable root of unity to recover all solutions. If it was not possible, z is set to zerosubgrouplistsubgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G, outputs the list of subgroups of G (of index bounded by bound, if not omitted), given as HNF left divisors of the SNF matrix corresponding to G. If flag=0 (default) and bnr is as output by bnrinit, gives only the subgroups for which the modulus is the conductorsubstGnGsubst(x,y,z): in expression x, replace the variable y by the expression zsubstpolsubstpol(x,y,z): in expression x, replace the polynomial y by the expression z, using remainder decomposition of x.substvecsubstvec(x,v,w): in expression x, make a best effort to replace the variables v1,...,vn by the expression w1,...,wnsumsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprsumaltsumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialssumdivGVEsumdiv(n,X,expr): sum of expression expr, X running over the divisors of nsuminfV=GEpsuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprsumnumV=GGEDGD0,L,psumnum(X=a,sig,expr,{tab},{flag=0}): numerical summation of expr from  X = ceiling(a) to +infinity. sig is either a scalar or a two-component vector coding the function's decrease rate at infinity. It is assumed that the scalar part of sig is to the right of all poles of expr. If present, tab must be initialized by sumnuminit. If flag is nonzero, assumes that conj(expr(z)) = expr(conj(z)).sumnumaltsumnumalt(X=a,sig,s,{tab},{flag=0}): numerical summation of (-1)^X s from X = ceiling(a) to +infinity. Note that the (-1)^X must not be included. sig is either a scalar or a two-component vector coded as in intnum, and the  scalar part is larger than all the real parts of the poles of s. Uses intnum, hence tab is as in intnum. If flag is nonzero, assumes that the function to  be summed satisfies conj(f(z))=f(conj(z)), and then up to twice faster.sumnuminitGD0,L,D1,L,psumnuminit(sig, {m=0}, {sgn=1}): initialize tables for numerical summation. sgn is 1 (in fact >= 0), the default, for sumnum (ordinary sums)  or -1 (in fact < 0) for sumnumalt (alternating sums). sig is as in sumnum and m is as in intnuminit.sumpossumpos(X=a,expr,{flag=0}): sum of positive series expr, the formal variable X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialstantan(x): tangent of xtanhtanh(x): hyperbolic tangent of xtaylorGnPtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmullerteichmuller(x): teichmuller character of p-adic number xthetatheta(q,z): Jacobi sine theta-functionthetanullkGLpthetanullk(q,k): k'th derivative at z=0 of theta(q,z)thuethue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a modulo units in the number field defined by P. If tnf was computed without assuming GRH (flag 1 in thueinit), the result is unconditionalthueinitthueinit(P,{flag=0}): initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero, certify the result unconditionnaly. Otherwise, assume GRH (much faster of course)tracetrace(x): trace of xtrapD"",r,DIDItrap({err}, {rec}, {seq}): try to execute seq, trapping error err (all of them if err ommitted); sequence rec is executed if the error occurs and is the result of the command. When seq is omitted, define rec as a default handler for error err (a break loop will be started if rec omitted). If rec is the empty string "" pop out the last default handlertruncatetruncate(x,{&e}): truncation of x; when x is a power series,take away the O(X^). If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitstypetype(x): return the type of the GEN x.untiluntil(a,seq): evaluate the expression sequence seq until a is nonzerovaluationvaluation(x,p): valuation of x with respect to pvariablevariable(x): main variable of object x. Gives p for p-adic x, error for scalarsvecextractvecextract(x,y,{z}): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designs columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in "1..10") or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)vecmaxvecmax(x): maximum of the elements of the vector/matrix xvecminvecmin(x): minimum of the elements of the vector/matrix xvecsortvecsort(x,{k},{flag=0}): sorts the vector of vectors (or matrix) x in ascending order, according to the value of its k-th component if k is not omitted. Binary digits of flag (if present) mean: 1: indirect sorting, return the permutation instead of the permuted vector, 2: sort using lexicographic order, 4: use descending instead of ascending ordervectorGDVDIvector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0svectorsmallvectorsmall(n,{X},{expr=0}): VECSMALL with n components of expression expr (X ranges from 1 to n) which must be small integers. By default, fill with 0svectorvvectorv(n,{X},{expr=0}): column vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0sweberweber(x,{flag=0}): One of Weber's f function of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that (j=(f^24-16)^3/f^24), 1: function f1(x)=eta(x/2)/eta(x) such that (j=(f1^24+16)^3/f2^24), 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that (j=(f2^24+16)^3/f2^24)whilewhile(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0writevss*write(filename,a): write the string expression a (same output as print) to filenamewrite1write1(filename,a): write the string expression a (same output as print1) to filenamewritebinvsDGwritebin(filename,{x}): write x as a binary object to file filename. If x is omitted, write all session variableswritetexwritetex(filename,a): write the string expression a (same format as print) to filename, in TeX formatzetazeta(s): Riemann zeta function at s with s a complex or a p-adic numberzetakzetak(nfz,s,{flag=0}): Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by zetakinit (NOT by nfinit) flag is optional, and can be 0: default, compute zetak, or non-zero: compute the lambdak function, i.e. with the gamma factorszetakinitzetakinit(x): compute number field information necessary to use zetak, where x is an irreducible polynomialzncoppersmithGGGDGzncoppersmith(P, N, X, {B=N}): finds all integers x0 with |x0| <= X such that  gcd(N, P(x0)) > B. X should be smaller than exp((log B)^2 / (deg(P) log N)).znlogznlog(x,g): g as output by znprimroot (modulo a prime). Return smallest non-negative n such that g^n = xznorderznorder(x,{o}): order of the integermod x in (Z/nZ)*. Optional o is assumed to be a multiple of the order.znprimrootznprimroot(n): returns a primitive root of n when it existsznstarznstar(n): 3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. eulerphi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorsuser interruptfloating point exception: bug in PARI or calling programbus error: bug in PARI or calling programsegmentation fault: bug in PARI or calling programunknown signalbroken pipebad object %Z"⍀"P
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Step 6
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MPQS: found factor = %ld whilst creating factor base
MPQS: sizing out of tune, FB size or tolerance
	too largeMPQSLPTMP%s/%sFRELFNEWLPRELLPNEWCOMBwrsMPQS: found %lu candidate%s

MPQS: passing the %3.1f%% sort point, time = %ld ms
MPQS: no factors found.

MPQS: time in Gauss and gcds = %ld ms

MPQS: restarting sieving ...

MPQS: giving up.
MPQS: found %ld factors =
	%Z%s
,MPQS: found factors = %Z
	and %Z
MPQS: found factor = %Z

MPQS: starting Gauss over F_2 on %ld relations
 and combiningMPQS: done sorting%s, time = %ld ms
MPQS: found %3.1f%% of the required relations
MPQS: found %ld full relations
MPQS: Net yield: %4.3g full relations per 100 candidates
MPQS:            %4.3g full relations per 100 polynomials
MPQS: %4.1f%% of the polynomials yielded no candidates
MPQS: next sort point at %3.1f%%
MPQS:   (%ld of these from partial relations)
MPQS: renamed file %s to %s
cannot rename file %s to %s
MPQS: split N whilst combining, time = %ld ms

MPQS: passing the %3.1f%% sort point
MPQS: Ran out of primes for A, giving up.
MPQS: starting main loop
MPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%%
MPQS: sieve threshold = %u
MPQS: sieving interval = [%ld, %ld]
MPQS: size of factor base = %ld
MPQS: striving for %ld relations
MPQS: coefficients A will be built from %ld primes each
MPQS: primes for A to be chosen near FB[%ld] = %ld
MPQS: smallest prime used for sieving FB[%ld] = %ld
MPQS: largest prime in FB = %ld
MPQS: bound for `large primes' = %ld
MPQS: sizing out of tune, FB too small or
	way too few primes in AMPQS: computing logarithm approximations for p_i in FB
MPQS: creating factor base and allocating arrays...
MPQS: number too big to be factored with MPQS,
	giving upMPQS: found multiplier %ld for N
MPQS: factoring number of %ld decimal digits
MPQS: number to factor N = %Z
manyseveralMPQS: factoring this number will take %s hours:
N = %ZMPQS: Gauss elimination will require more than
	128MBy of memory	(estimated memory needed: %4.1fMBy)
MPQS: kN = %Z
MPQS: kN has %ld decimal digits
,<%lu>,%lu,%lu...] Wait a second --
]
MPQS: FB [-1,2MPQS: precomputing auxiliary primes up to %ld
error whilst writing to file %sMPQS: done sorting one file.
MQPS: short of space -- another buffer for sorting
MPQS: relations file truncated?!
MQPS: line wrap -- another buffer for sorting
error whilst flushing file %sMPQS: chose Q_%ld(x) = %Z x^2 %c %Z x + C
 %lu %lu%s @ %s :%s
%s :%s
 
 %ld %ld : 0
MPQS: combining
    {%ld @ %s : %s}
  * {%ld @ %s : %s}
 == {%s}
 1 1MPQS: combined %ld full relation%s
MPQS: Gauss done: kernel has rank %ld, taking gcds...
, looking for more...MPQS: got %ld factors%s
comp.unknown	packaging %ld: %Z ^%ld (%s)
MPQS: wrapping up vector of %ld factors
[3]: mpqs_solve_linear_systemMPQS: resplitting a factor after %ld kernel vectors
MPQS: splitting N after %ld kernel vector%s
MPQS: got two factors, looking for more...
MPQS: X^2 - Y^2 != 0 mod N
	index i = %ld
MPQS: wrong relation found after Gauss[2]: mpqs_solve_linear_systemMPQS (relation is a nonsquare)[1]: mpqs_solve_linear_systemFREL file truncated?!cannot seek FREL fileMPQS: no solutions found from linear system solver\\ KERNEL COMPUTED BY MPQS
KERNEL=
[1, 0, 01; \\ MATRIX READ BY MPQS
FREL=MPQS: chose primes for A FB[%ld]=%ld%sMPQS: new bit pattern for primes for A: 0x%lX
MPQS: wrapping, more primes for A now chosen near FB[%ld] = %ld
ftell error on full relations fileshorterlongerMPQS: full relations file %s than expectedMPQS panickingMPQS: decomposed a square
cube5th power7th powerMPQS: decomposed a %s
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nfissplitnffactorpolynomial variable must have highest priority in nffactornumber of factor(s) found: %ld
squarefree test
Entering nffactor:
nfsqffUsing Trager's method
Total Time: %ld
===========
splitting mod %Zbound computationfactorroot  1) T_2 bound for %s: %Z
  2) Conversion from T_2 --> | |^2 bound : %Z
  3) Final bound: %Z
Mignotte bound: %Z
Beauzamy bound: %Z
choice of a prime idealPrime ideal chosen: %Z
incorrect variables in rnfcharpolynf_factor_boundfor this exponent, GSmin = %Z
Time reduction: %ld
exponent: %ld
Naive recombinationHensel liftrootsfactors%3ld %s at prime
  %Z
Time: %ld

### K = %d, %Z combinations

to find factor %Zremaining modular factor(s): %ld
*@|.nf_LLL_cmbf* Time LLL: %ld
* Time Check Factor: %ld
for this traceLLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld

LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)
... lifted (avma - bot = %lu)
... mod p^k (avma - bot = %lu)
nf_LLL_cmbf: checking factor %ld (avma - bot = %lu)
special_pivot output:
%Z
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polrelnum = %Z
It's not a square...
Checking the square-root of the Stark unit...
zetavalues = %Z
N0 in QuickPol: %ld 
Compute Wyquadhilbertrealnew precision: %ld
FindModulusCompute Cl(k)class field not totally real in bnrstarkincorrect subgrp in bnrstarkbase field not totally real in bnrstarkmain variable in bnrstark must not be xno non-trivial character in bnrL1incorrect subgroup in bnrL1bnrL1the ground field must be distinct from QNo, we're done!
Modulus = %Z and subgroup = %Z
Trying to find another modulus...
Trying modulus = %Z and subgroup = %Z
 %ldLooking for a modulus of norm: ArtinNumberconductor too large in ArtinNumber* Root Number: cond. no %ld/%ld (%ld chars)
%ld RecCoeff (cf = %ld, B = %Z)
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	Init: N0 = %ld
Not enough precomputed primes (need all p <= %ld)S&TCplxModuluscpl = 2^%ld
diff(CHI) = %ZRecCoeffRecCoeff3: no solution found!
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Z = %Z
Y = %Z

ns = %ld
changing f(x): p divides disc(g)
embedding = %Z
candidate = %Z
coeff too big for pol g(x)
pol. found = %Z
d-1 test failed
delta[%ld] = %Z

Subfields of degree %ld: %Z

* Look for subfields of degree %ld

coeff too big for embedding
lifting embedding mod p^k = %Z^%ld
Chosen prime: p = %ld
p = %ld,	lcm = %ld,	orbits: %Z
sorry, too many block systems in nfsubfieldsf = %Z
p = %Z, lift to p^%ld
2 * Hadamard bound * ind = %Z
2 * M = %Z
Entering compute_data()


***** Leaving subfields


***** Entering subfields

pol = %Z
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thueinitprec = %d
invalid polynomial in thue (need deg>2)thue  c6  = %Z
  c8  = %Z
  c11 = %Z
  c15 = %Z
  - norm sol. no %ld/%ld
get_emb* real root no %ld/%ld
  c10 = %Z
  c13 = %Z
x1 -> %Z
x2 -> %Z
c14 = %Z
Non trivial conditional class group.
  *** May miss solutions of the norm equationAll solutions are <= %Z
not a tnf in thueexpected an integer in bnfisintnorm%Z eliminated because of sign
bnfisintnormlooking for a fundamental unit of norm -1
invalid polynomial in thue (need n>2)inithueepsilon_3 -> %Z
c1 = %Z
c2 = %Z
Indice <= %Z
non-monic polynomial in thueNot enough precision in thueChecking (\pm %Z, \pm %Z)
overflow in real shiftSmallSols* Checking for small solutions
Semirat. reduction: B0 -> %Z
thue (totally rational case)CF failed. Increasing kappa
    B0 -> %Z
  Entering CF...
  errdelta = %Z
LLL failed. Increasing kappa
Semirat. reduction: B0 -> %Z x <= %Z
  Entering LLL...
sol = %Z
Partial = %Z
gcd f_P  does not divide n_p
  B0  = %Z
  Baker = %Z
C (bitsize) : %d
LLL_First_Pass successful !!
B0 -> %Z
x <= %Z
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