PARI-GP Reference Card
(PARI-GP version 2.6.1)
Note: optional arguments are surrounded by braces {}.
To start the calculator, type its name in the terminal: gp
To exit gp, type quit, \q, or <C-D> at prompt.
Help
describe function ?function
extended description ??keyword
list of relevant help topics ???pattern
Input/Output
previous result, the result before %, %‘, %‘‘, etc.
n-th result since startup %n
separate multiple statements on line ;
extend statement on additional lines \
extend statements on several lines {seq1; seq2;}
comment /* . . . */
one-line comment, rest of line ignored \\ . . .
Metacommands & Defaults
set default d to val default({d}, {val}, {flag})
toggle timer on/off #
print time for last result ##
print defaults \d
set debug level to n \g n
set memory debug level to n \gm n
set output mode (raw=0, default=1) \o n
set n significant digits \p n
set n terms in series \ps n
quit GP \q
print the list of PARI types \t
print the list of user-defined functions \u
read file into GP \r filename
Debugger / break loop
get out of break loop break or <C-D>
go up n frames dbg up({n})
examine object o dbg x(o)
PARI Types & Input Formats
t INT/t REAL. Integers, Reals ±n, ±n.ddd
t INTMOD. Integers modulo m Mod(n, m)
t FRAC. Rational Numbers n/m
t FFELT. Elt in finite field Fq ffgen(q)
t COMPLEX. Complex Numbers x + y ∗ I
t PADIC. p-adic Numbers x + O(p^k)
t QUAD. Quadratic Numbers x + y ∗ quadgen(D)
t POLMOD. Polynomials modulo g Mod(f, g)
t POL. Polynomials a ∗ x^n + · · · + b
t SER. Power Series f + O(x^k)
t QFI/t QFR. Imag/Real bin. quad. forms Qfb(a, b, c, {d})
t RFRAC. Rational Functions f/g
t VEC/t COL. Row/Column Vectors [x, y, z], [x, y, z]~
t MAT. Matrices [x, y;z, t;u, v]
t LIST. Lists List([x, y, z])
t STR. Strings "abc"
Reserved Variable Names
π = 3.14 . . ., γ = 0.57 . . ., C = 0.91 . . . Pi, Euler, Catalan
square root of −1 I
big-oh notation O
c 2013 Karim Belabas. Permissions on back. v2.27
Information about an Object
PARI type of object x type(x)
length of x / size of x in memory #x, sizebyte(x)
real or p-adic precision of x precision(x), padicprec
Operators
basic operations +, - , *, /, ^
i=i+1, i=i-1, i=i*j, . . . i++, i--, i*=j,. . .
euclidean quotient, remainder x\/y, x\y, x%y, divrem(x, y)
shift x left or right n bits x<<n, x>>n or shift(x, ±n)
comparison operators <=, <, >=, >, ==, !=, ===, lex, cmp
boolean operators (or, and, not) ||, &&, !
bit operations bitand, bitneg, bitor, bitxor
sign of x = −1, 0, 1 sign(x)
maximum/minimum of x and y max, min(x, y)
integer or real factorial of x x! or factorial(x)
derivative of f w.r.t. x f’
apply differential operator diffop
restore x as a formal variable x=’x
simultaneous assignment x ← v1, y ← v2 [x,y] = v
Select Components
n-th component of x component(x, n)
n-th component of vector/list x x[n]
components a, a + 1, . . . , b of vector x x[a..b]
(m, n)-th component of matrix x x[m, n]
row m or column n of matrix x x[m,], x[, n]
numerator/denominator of x numerator(x), denominator
Conversions
to vector, matrix, set, list, string Col/Vec,Mat,Set,List,Str
create PARI object (x mod y) Mod(x, y)
make x a polynomial of v Pol(x, {v})
as Pol/Vec, starting with constant term Polrev, Vecrev
make x a power series of v Ser(x, {v})
string from bytes / from format+args Strchr, Strprintf
convert x to simplest possible type simplify(x)
object x with precision n precision(x, n)
Conjugates and Lifts
conjugate of a number x conj(x)
conjugate vector of algebraic number x conjvec(x)
norm of x, product with conjugate norm(x)
square of L2 norm of vector x norml2(x)
lift of x from Mods lift, centerlift(x)
Lists, Sets & Sorting
sort x by k-th component vecsort(x, {k}, {fl = 0})
min. m of x (m = x[i]), max. vecmin(x, {&i}), vecmax
does y belong to x, sorted wrt. f vecsearch(x, y, {f})
Sets (= row vector of strings with strictly increasing entries)
intersection of sets x and y setintersect(x, y)
set of elements in x not belonging to y setminus(x, y)
union of sets x and y setunion(x, y)
does y belong to the set x setsearch(x, y, {flag})
is x a set ? setisset(x)
Lists. create empty list: L = List()
append x to list L listput(L, x, {i})
remove i-th component from list L listpop(L, {i})
insert x in list L at position i listinsert(L, x, i)
sort the list L in place listsort(L, {flag})
Programming
Functions and closures
fun(vars) = my(local vars); seq
fun = (vars) -> my(local vars); seq
Control Statements (X: formal parameter in expression seq)
eval. seq for a ≤ X ≤ b for(X = a, b, seq)
eval. seq for X dividing n fordiv(n, X, seq)
eval. seq for primes a ≤ X ≤ b forprime(X = a, b, seq)
eval. seq for a ≤ X ≤ b stepping s forstep(X = a, b, s, seq)
multivariable for forvec(X = v, seq)
loop over partitions of n forpart(p =n seq)
loop over vectors v, q(v) ≤ B, q > 0 forqfvec(v, q, b, seq)
loop over subgrps H of abelian grp G forsubgroup(H = G)
evaluate seq until a = 0 until(a, seq)
while a = 0, evaluate seq while(a, seq)
exit n innermost enclosing loops break({n})
start new iteration of n-th enclosing loop next({n})
return x from current subroutine return({x})
raise an exception error()
if a = 0, evaluate seq1, else seq2 if(a, {seq1}, {seq2})
try seq1, evaluate seq2 on error iferr(seq1, E, seq2)
select from v according to f select(f, v)
apply f to all entries in v apply(f, v)
Input/Output
print with/without \n, TEX format print, print1, printtex
formatted printing printf()
write args to file write, write1, writetex(file, args)
write x in binary format writebin(file, x)
read file into GP read({file})
read file, return as vector of lines readvec({file})
read a string from keyboard input()
Interface with User and System
allocates a new stack of s bytes allocatemem({s})
alias old to new alias(new, old)
install function from library install(f, code, {gpf }, {lib})
execute system command a system(a)
as above, feed result to GP extern(a)
as above, return GP string externstr(a)
get $VAR from environment getenv("VAR")
measure time in ms. gettime()
timeout command after s seconds alarm(s, expr)
Iterations, Sums & Products
numerical integration intnum(X = a, b, expr, {flag})
sum expr over divisors of n sumdiv(n, X, expr)
sumdiv, with expr multiplicative sumdivmult(n, X, expr)
sum X = a to X = b, initialized at x sum(X = a, b, expr, {x})
sum of series expr suminf(X = a, expr)
sum of alternating/positive series sumalt, sumpos
sum of series using intnum sumnum
product a ≤ X ≤ b, initialized at x prod(X = a, b, expr, {x})
product over primes a ≤ X ≤ b prodeuler(X = a, b, expr)
infinite product a ≤ X ≤ ∞ prodinf(X = a, expr)
real root of expr between a and b solve(X = a, b, expr)
Random Numbers
random integer/prime in [0, N[ random(N), randomprime
get/set random seed getrand, setrand(s)
Vectors & Matrices
dimensions of matrix x matsize(x)
concatenation of x and y concat(x, {y})
extract components of x vecextract(x, y, {z})
transpose of vector or matrix x mattranspose(x) or x~
adjoint of the matrix x matadjoint(x)
eigenvectors/values of matrix x mateigen(x)
characteristic/minimal polynomial of x charpoly(x), minpoly
trace/determinant of matrix x trace(x), matdet
Frobenius form of x matfrobenius(x)
QR decomposition matqr(x)
Constructors & Special Matrices
row vec. of expr eval’ed at 1 ≤ i ≤ n vector(n, {i}, {expr})
col. vec. of expr eval’ed at 1 ≤ i ≤ n vectorv(n, {i}, {expr})
matrix 1 ≤ i ≤ m, 1 ≤ j ≤ n matrix(m, n, {i}, {j}, {expr})
define matrix by blocks matconcat(B)
diagonal matrix with diagonal x matdiagonal(x)
n × n identity matrix matid(n)
Hessenberg form of square matrix x mathess(x)
n × n Hilbert matrix Hij = (i + j − 1)−1 mathilbert(n)
companion matrix to polynomial x matcompanion(x)
Sylvester matrix of x polsylvestermatrix(x)
Gaussian elimination
kernel of matrix x matker(x, {flag})
intersection of column spaces of x and y matintersect(x, y)
solve M ∗ X = B (M invertible) matsolve(M, B)
as solve, modulo D (col. vector) matsolvemod(M, D, B)
one sol of M ∗ X = B matinverseimage(M, B)
basis for image of matrix x matimage(x)
supplement columns of x to get basis matsupplement(x)
rows, cols to extract invertible matrix matindexrank(x)
rank of the matrix x matrank(x)
Lattices & Quadratic Forms
upper triangular Hermite Normal Form mathnf(x)
HNF of x where d is a multiple of det(x) mathnfmod(x, d)
elementary divisors of x matsnf(x)
LLL-algorithm applied to columns of x qflll(x, {flag})
like qflll, x is Gram matrix of lattice qflllgram(x, {flag})
LLL-reduced basis for kernel of x matkerint(x)
Z-lattice ←→ Q-vector space matrixqz(x, p)
signature of quad form ty ∗ x ∗ y qfsign(x)
decomp into squares of ty ∗ x ∗ y qfgaussred(x)
eigenvals/eigenvecs for real symmetric x qfjacobi(x)
find up to m sols of ty ∗ x ∗ y ≤ b qfminim(x, b, m)
perfection rank of x qfperfection(x)
v, v[i] :=number of sols of ty ∗ x ∗ y = i qfrep(x, B, {flag})
automorphism group of q qfauto(q)
find isomorphism between q and Q qfisom(q, Q)
Formal & p-adic Series
truncate power series or p-adic number truncate(x)
valuation of x at p valuation(x, p)
Dirichlet and Power Series
Taylor expansion around 0 of f w.r.t. x taylor(f, x)
akbktk from aktk and bktk serconvol(a, b)
f = aktk from (ak/k!)tk serlaplace(f)
reverse power series F so F(f(x)) = x serreverse(f)
Dirichlet series multiplication / division dirmul, dirdiv(x, y)
Dirichlet Euler product (b terms) direuler(p = a, b, expr)
PARI-GP Reference Card
(PARI-GP version 2.6.1)
Polynomials & Rational Functions
degree of f poldegree(f)
coeff. of degree n of f, leading coeff. polcoeff(f, n), pollead
gcd of coefficients of f content(f)
replace x by y subst(f, x, y)
evaluate f replacing vars by their value eval(f)
replace polynomial expr. T(x) by y in f substpol(f, T, y)
replace x1, . . . , xn by y1, . . . , yn in f substvec(f, x, y)
discriminant of polynomial f poldisc(f)
resultant R = Resv(f, g) polresultant(f, g, {v})
[u, v, R], xu + yv = Resv(f, g) polresultantext(x, y, {v})
derivative of f w.r.t. x deriv(f, {x})
formal integral of f w.r.t. x intformal(f, {x})
formal sum of f w.r.t. x sumformal(f, {x})
reciprocal poly xdeg f f(1/x) polrecip(f)
interpol. pol. eval. at a polinterpolate(X, {Y }, {a}, {&e})
initialize t for Thue equation solver thueinit(f)
solve Thue equation f(x, y) = a thue(t, a, {sol})
Roots and Factorization
number of real roots of f, a < x ≤ b polsturm(f, {a}, {b})
complex roots of f polroots(f)
symmetric powers of roots of f up to n polsym(f, n)
factor f factor(f, {lim})
factor f mod p / roots factormod(f, p), polrootsmod
factor f over Fpa / roots factorff(f, p, a), polrootsff
factor f over Qp / roots factorpadic(f, p, r), polrootspadic
find irreducible T ∈ Fp[x], deg T = n ffinit(p, n, {x})
#{monic irred. T ∈ Fq[x], deg T = n} ffnbirred(q, n)
p-adic root of f cong. to a mod p padicappr(f, a)
Newton polygon of f for prime p newtonpoly(f, p)
extensions of Qp of degree N padicfields(p, N)
Special Polynomials
n-th cyclotomic polynomial in var. v polcyclo(n, {v})
d-th degree subfield of Q(ζn) polsubcyclo(n, d, {v})
Pn, Tn/Un, Hn pollegendre, polchebyshev, polhermite
Transcendental and p-adic Functions
real, imaginary part of x real(x), imag(x)
absolute value, argument of x abs(x), arg(x)
square/nth root of x sqrt(x), sqrtn(x, n, {&z})
trig functions sin, cos, tan, cotan
inverse trig functions asin, acos, atan
hyperbolic functions sinh, cosh, tanh
inverse hyperbolic functions asinh, acosh, atanh
exponential / natural log of x exp, log
Euler Γ function, log Γ, Γ /Γ gamma, lngamma, psi
incomplete gamma function (y = Γ(s)) incgam(s, x, {y})
exponential integral ∞
x e−t/t dt eint1(x)
error function 2/
√
π ∞
x e−t2
dt erfc(x)
dilogarithm of x dilog(x)
m-th polylogarithm of x polylog(m, x, {flag})
U-confluent hypergeometric function hyperu(a, b, u)
Bessel Jn(x), Jn+1/2(x) besselj(n, x), besseljh(n, x)
Bessel Iν, Kν, H1
ν , H2
ν , Nν (bessel)i, k, h1, h2, n
Lambert W: x s.t. xex = y lambertw(y)
Teichmuller character of p-adic x teichmuller(x)
Elementary Arithmetic Functions
vector of binary digits of |x| binary(x)
bit number n of integer x bittest(x, n)
Hamming weight of integer x hammingweight(x)
ceiling/floor/fractional part ceil, floor, frac
round x to nearest integer round(x, {&e})
truncate x truncate(x, {&e})
gcd/LCM of x and y gcd(x, y), lcm(x, y)
gcd of entries of a vector/matrix content(x)
Primes and Factorization
add primes in v to prime table addprimes(v)
Chebyshev π(x), n-th prime pn primepi(x), prime(n)
vector of first n primes primes(n)
smallest prime ≥ x nextprime(x)
largest prime ≤ x precprime(x)
factorization of x factor(x, {lim})
n = df2, d squarefree/fundamental core(n, {fl}), coredisc
recover x from its factorization factorback(f, {e})
Divisors
number of prime divisors ω(n) / Ω(n) omega(n), bigomega
divisors of n / number of divisors τ(n) divisors(n), numdiv
sum of (k-th powers of) divisors of n sigma(n, {k})
Special Functions and Numbers
binomial coefficient x
y binomial(x, y)
Bernoulli number Bn as real/rational bernreal(n), bernfrac
Bernoulli polynomial Bn(x) bernpol(n, {x})
n-th Fibonacci number fibonacci(n)
Stirling numbers s(n, k) and S(n, k) stirling(n, k, {flag})
number of partitions of n numbpart(n)
M¨obius µ-function moebius(x)
Hilbert symbol of x and y (at p) hilbert(x, y, {p})
Kronecker-Legendre symbol ( x
y ) kronecker(x, y)
Dedekind sum s(h, k) sumdedekind(h, k)
Multiplicative groups (Z/NZ)∗, F∗
q
Euler φ-function eulerphi(x)
multiplicative order of x (divides o) znorder(x, {o}), fforder
primitive root mod q / x.mod znprimroot(q), ffprimroot(x)
structure of (Z/nZ)∗ znstar(n)
discrete logarithm of x in base g znlog(x, g, {o}), fflog
Miscellaneous
integer square / n-th root of x sqrtint(x), sqrtnint(x, n)
solve z ≡ x and z ≡ y chinese(x, y)
minimal u, v so xu + yv = gcd(x, y) gcdext(x, y)
continued fraction of x contfrac(x, {b}, {lmax})
last convergent of continued fraction x contfracpnqn(x)
rational approximation to x bestappr(x, k), bestapprPade
True-False Tests
is x the disc. of a quadratic field? isfundamental(x)
is x a prime? isprime(x)
is x a strong pseudo-prime? ispseudoprime(x)
is x square-free? issquarefree(x)
is x a square? issquare(x, {&n})
is x a perfect power? ispower(x, {k}, {&n})
is pol irreducible? polisirreducible(pol)
Based on an earlier version by Joseph H. Silverman
September 2013 v2.27. Copyright c 2013 K. Belabas
Permission is granted to make and distribute copies of this card provided
the copyright and this permission notice are preserved on all copies.
Send comments and corrections to Karim.Belabas@math.u-bordeaux.fr
PARI-GP Reference Card (2)
(PARI-GP version 2.6.1)
Elliptic Curves
Elliptic curve initially given by 5-tuple v =[a1, a2, a3, a4, a6].
Initialize ell struct E = ellinit(v, {Domain})
Points are [x,y], the origin is [0]. Struct members accessed
as E.member:
• All domains: E.a1,a2,a3,a4,a6, b2,b4,b6,b8, c4,c6, disc, j
• E defined over R or C
x-coords. of points of order 2 E.roots
periods / quasi-periods E.omega,E.eta
volume of complex lattice E.area
• E defined over Qp
residual characteristic E.p
If |j|p > 1: Tate’s [u2, u, q, [a, b]] E.tate
• E defined over Fq
characteristic E.p
#E(Fq)/cyclic structure/generators E.no, E.cyc, E.gen
• E defined over Q
generators of E(Q) (require elldata) E.gen
[a1, a2, a3, a4, a6] from j-invariant ellfromj(j)
change curve E using v =[u, r, s, t] ellchangecurve(E, v)
change point z using v =[u, r, s, t] ellchangepoint(z, v)
add points P + Q / P − Q elladd(E, P, Q), ellsub
negate point ellneg(E, P)
compute n · z ellmul(E, z, n)
n-division polynomial fn(x) elldivpol(E, n, {x})
check if z is on E ellisoncurve(E, z)
order of torsion point z ellorder(E, z)
y-coordinates of point(s) for x ellordinate(E, x)
point [℘(z), ℘ (z)] corresp. to z ellztopoint(E, z)
complex z such that p = [℘(z), ℘ (z)] ellpointtoz(E, p)
Curves over finite fields, Pairings
random point on E random(E)
#E(Fq) ellcard(E)
structure Z/d1Z × Z/d2Z of E(Fq) ellgroup(E)
Weil pairing of m-torsion pts x, y ellweilpairing(E, x, y, m)
Tate pairing of x, y; x m-torsion elltatepairing(E, x, y, m)
Discrete log, find n s.t. P = [n]Q elllog(E, P, Q, {ord})
Curves over Q and the L-function
canonical bilinear form taken at z1, z2 ellbil(E, z1, z2)
canonical height of z ellheight(E, z, {flag})
height regulator matrix for pts in x ellheightmatrix(E, x)
cond, min mod, Tamagawa num [N, v, c] ellglobalred(E)
reduction of y2 + Qy = P (genus 2) genus2red(Q, P, {p})
Kodaira type of p-fiber of E elllocalred(E, p)
minimal model of E/Q ellminimalmodel(E, {&v})
p-th coeff ap of L-function, p prime ellap(E, p)
k-th coeff ak of L-function ellak(E, k)
vector of first n ak’s in L-function ellan(E, n)
L(E, s) elllseries(E, s)
L(r)(E, 1) ellL1(E, r)
return a Heegner point on E of rank 1 ellheegner(E)
order of vanishing at 1 ellanalyticrank(E, {eps})
root number for L(E, .) at p ellrootno(E, {p})
torsion subgroup with generators elltors(E)
modular parametrization of E elltaniyama(E)
c 2013 Karim Belabas. Permissions on back. v2.27
Elldata package, Cremona’s database:
db code ↔ [conductor, class, index] ellconvertname(s)
generators of Mordell-Weil group ellgenerators(E)
look up E in database ellidentify(E)
all curves matching criterion ellsearch(N)
loop over curves with cond. from a to b forell(E, a, b, seq)
Elliptic & Modular Functions
w = [ω1, ω2] or ell struct (E.omega), τ = ω1/ω2.
arithmetic-geometric mean agm(x, y)
elliptic j-function 1/q + 744 + · · · ellj(x)
Weierstrass σ/℘/ζ function ellsigma(w, z), ellwp, ellzeta
periods/quasi-periods ellperiods(E, {flag}), elleta(w)
(2iπ/ω2)kEk(τ) elleisnum(w, k, {flag})
modified Dedekind η func. (1 − qn) eta(x, {flag})
Jacobi sine theta function theta(q, z)
k-th derivative at z=0 of theta(q, z) thetanullk(q, k)
Weber’s f functions weber(x, {flag})
Riemann’s zeta ζ(s) = n−s zeta(s)
Binary Quadratic Forms
create ax2 + bxy + cy2 (distance d) Qfb(a, b, c, {d})
reduce x (s =
√
D, l = s ) qfbred(x, {flag}, {D}, {l}, {s})
composition of forms x*y or qfbnucomp(x, y, l)
n-th power of form x^n or qfbnupow(x, n)
composition without reduction qfbcompraw(x, y)
n-th power without reduction qfbpowraw(x, n)
prime form of disc. x above prime p qfbprimeform(x, p)
class number of disc. x qfbclassno(x)
Hurwitz class number of disc. x qfbhclassno(x)
Solve Q(x, y) = p in integers, p prime qfbsolve(Q, p)
Quadratic Fields
quadratic number ω =
√
x or (1 +
√
x)/2 quadgen(x)
minimal polynomial of ω quadpoly(x)
discriminant of Q(
√
D) quaddisc(x)
regulator of real quadratic field quadregulator(x)
fundamental unit in real Q(x) quadunit(x)
class group of Q(
√
D) quadclassunit(D, {flag}, {t})
Hilbert class field of Q(
√
D) quadhilbert(D, {flag})
ray class field modulo f of Q(
√
D) quadray(D, f, {flag})
General Number Fields: Initializations
A number field K is given by a monic irreducible f ∈ Z[X].
init number field structure nf nfinit(f, {flag})
nf members:
polynomial defining nf , f(θ) = 0 nf .pol
number of real/complex places nf .r1/r2/sign
discriminant of nf nf .disc
T2 matrix nf .t2
vector of roots of f nf .roots
integral basis of ZK as powers of θ nf .zk
different nf .diff
codifferent nf .codiff
index nf .index
recompute nf using current precision nfnewprec(nf)
init relative rnf given by g = 0 over K rnfinit(nf , g)
initbnf structure bnfinit(f, {flag})
bnf members: same as nf , plus
underlying nf bnf .nf
classgroup bnf .clgp
regulator bnf .reg
fundamental units bnf .fu
torsion units bnf .tu
compute a bnf from small bnf bnfinit(sbnf )
add S-class group and units, yield bnf s bnfsunit(nf , S)
init class field structure bnr bnrinit(bnf , m, {flag})
bnr members: same as bnf , plus
underlying bnf bnr.bnf
big ideal structure bnr.bid
modulus bnr.mod
structure of (ZK/m)∗ bnr.zkst
Basic Number Field Arithmetic (nf)
Elements are t INT, t FRAC, t POL, t POLMOD, or t COL (on integral
basis nf .zk). Basic operations (prefix nfelt): (nfelt)add,
mul, pow, div, diveuc, mod, divrem, val, trace, norm
express x on integer basis nfalgtobasis(nf , x)
express element x as a polmod nfbasistoalg(nf , x)
reverse polmod a = A(X) mod T(X) modreverse(a)
integral basis of field def. by f = 0 nfbasis(f)
field discriminant of field f = 0 nfdisc(f)
smallest poly defining f = 0 (slow) polredabs(f, {flag})
small poly defining f = 0 (fast) polredbest(f, {flag})
are fields f = 0 and g = 0 isomorphic? nfisisom(f, g)
is field f = 0 a subfield of g = 0? nfisincl(f, g)
compositum of f = 0, g = 0 polcompositum(f, g, {flag})
subfields (of degree d) of nf nfsubfields(nf , {d})
roots of unity in nf nfrootsof1(nf )
roots of g belonging to nf nfroots({nf }, g)
factor g in nf nffactor(nf , g)
factor g mod prime pr in nf nffactormod(nf , g, pr)
conjugates of a root θ of nf nfgaloisconj(nf , {flag})
apply Galois automorphism s to x nfgaloisapply(nf , s, x)
quadratic Hilbert symbol (at p) nfhilbert(nf , a, b, {p})
Linear and algebraic relations
poly of degree ≤ k with root x ∈ C algdep(x, k)
alg. dep. with pol. coeffs for series s seralgdep(s, x, y)
small linear rel. on coords of vector x lindep(x)
Dedekind Zeta Function ζK, Hecke L series
ζK as Dirichlet series, N(I) < b dirzetak(nf , b)
init nfz for field f = 0 zetakinit(f)
compute ζK(s) zetak(nfz, s, {flag})
Artin root number of K bnrrootnumber(bnr, chi, {flag})
L(1, χ), for all χ trivial on H bnrL1(bnr, {H}, {flag})
Class Groups & Units (bnf, bnr)
a1, {a2}, {a3} usually bnr, subgp or bnf , module, {subgp}
remove GRH assumption from bnf bnfcertify(bnf )
expo. of ideal x on class gp bnfisprincipal(bnf , x, {flag})
expo. of ideal x on ray class gp bnrisprincipal(bnr, x, {flag})
expo. of x on fund. units bnfisunit(bnf , x)
as above for S-units bnfissunit(bnfs, x)
signs of real embeddings of bnf .fu bnfsignunit(bnf )
narrow class group bnfnarrow(bnf )
Class Field Theory
ray class number for mod. m bnrclassno(bnf , m)
discriminant of class field ext bnrdisc(a1, {a2}, {a3})
ray class numbers, l list of mods bnrclassnolist(bnf , l)
discriminants of class fields bnrdisclist(bnf , l, {arch}, {flag})
decode output from bnrdisclist bnfdecodemodule(nf , fa)
is modulus the conductor? bnrisconductor(a1, {a2}, {a3})
conductor of character chi bnrconductorofchar(bnr, chi)
conductor of extension bnrconductor(a1, {a2}, {a3}, {flag})
conductor of extension def. by g rnfconductor(bnf , g)
Artin group of ext. def’d by g rnfnormgroup(bnr, g)
subgroups of bnr, index <= b subgrouplist(bnr, b, {flag})
rel. eq. for class field def’d by sub rnfkummer(bnr, sub, {d})
same, using Stark units (real field) bnrstark(bnr, sub, {flag})
Ideals: elements, primes, or matrix of generators in HNF
is id an ideal in nf ? nfisideal(nf , id)
is x principal in bnf ? bnfisprincipal(bnf , x)
give [a, b], s.t. aZK + bZK = x idealtwoelt(nf , x, {a})
put ideal a (aZK + bZK) in HNF form idealhnf(nf , a, {b})
norm of ideal x idealnorm(nf , x)
minimum of ideal x (direction v) idealmin(nf , x, v)
LLL-reduce the ideal x (direction v) idealred(nf , x, {v})
Ideal Operations
add ideals x and y idealadd(nf , x, y)
multiply ideals x and y idealmul(nf , x, y, {flag})
intersection of ideals x and y idealintersect(nf , x, y, {flag})
n-th power of ideal x idealpow(nf , x, n, {flag})
inverse of ideal x idealinv(nf , x)
divide ideal x by y idealdiv(nf , x, y, {flag})
Find (a, b) ∈ x × y, a + b = 1 idealaddtoone(nf , x, {y})
coprime integral A, B such that x = A/B idealnumden(nf , x)
Primes and Multiplicative Structure
factor ideal x in nf idealfactor(nf , x)
expand ideal factorization in nf idealfactorback(nf, f, e)
decomposition of prime p in nf idealprimedec(nf , p)
valuation of x at prime ideal pr idealval(nf , x, pr)
weak approximation theorem in nf idealchinese(nf , x, y)
give bid =structure of (ZK/id)∗ idealstar(nf , id, {flag})
discrete log of x in (ZK/bid)∗ ideallog(nf , x, bid)
idealstar of all ideals of norm ≤ b ideallist(nf , b, {flag})
add Archimedean places ideallistarch(nf , b, {ar}, {flag})
init prmod structure nfmodprinit(nf , pr)
kernel of matrix M in (ZK/pr)∗ nfkermodpr(nf , M, prmod)
solve Mx = B in (ZK/pr)∗ nfsolvemodpr(nf , M, B, prmod)
Galois theory over Q
Galois group of field Q[x]/(f) polgalois(f)
initializes a Galois group structure G galoisinit(pol, {den})
action of p in nfgaloisconj form galoispermtopol(G, {p})
identify as abstract group galoisidentify(G)
export a group for GAP/MAGMA galoisexport(G, {flag})
subgroups of the Galois group G galoissubgroups(G)
is subgroup H normal? galoisisnormal(G, H)
subfields from subgroups galoissubfields(G, {flag}, {v})
fixed field galoisfixedfield(G, perm, {flag}, {v})
Frobenius at maximal ideal P idealfrobenius(nf , G, P)
ramification groups at P idealramgroups(nf , G, P)
PARI-GP Reference Card (2)
(PARI-GP version 2.6.1)
is G abelian? galoisisabelian(G, {flag})
abelian number fields/Q galoissubcyclo(N,H,{flag},{v})
query the galpol package galoisgetpol(a,b,{s})
Relative Number Fields (rnf)
Extension L/K is defined by T ∈ K[x].
absolute equation of L rnfequation(nf , T, {flag})
is L/K abelian? rnfisabelian(nf , T)
relative nfalgtobasis rnfalgtobasis(rnf , x)
relative nfbasistoalg rnfbasistoalg(rnf , x)
relative idealhnf rnfidealhnf(rnf , x)
relative idealmul rnfidealmul(rnf , x, y)
relative idealtwoelt rnfidealtwoelt(rnf , x)
Lifts and Push-downs
absolute → relative repres. for x rnfeltabstorel(rnf , x)
relative → absolute repres. for x rnfeltreltoabs(rnf , x)
lift x to the relative field rnfeltup(rnf , x)
push x down to the base field rnfeltdown(rnf , x)
idem for x ideal: (rnfideal)reltoabs, abstorel, up, down
Norms
absolute norm of ideal x rnfidealnormabs(rnf , x)
relative norm of ideal x rnfidealnormrel(rnf , x)
solutions of NK/Q(y) = x ∈ Z bnfisintnorm(bnf , x)
is x ∈ Q a norm from K? bnfisnorm(bnf , x, {flag})
initialize T for norm eq. solver rnfisnorminit(K, pol, {flag})
is a ∈ K a norm from L? rnfisnorm(T, a, {flag})
Maximal order ZL as a ZK-module
relative polred rnfpolred(nf , T)
relative polredabs rnfpolredabs(nf , T)
characteristic poly. of a mod T rnfcharpoly(nf , T, a, {v})
relative Dedekind criterion, prime pr rnfdedekind(nf , T, pr)
discriminant of relative extension rnfdisc(nf , T)
pseudo-basis of ZL rnfpseudobasis(nf , T)
General ZK-modules: M = [matrix, vec. of ideals] ⊂ L
relative HNF / SNF nfhnf(nf , M), nfsnf
reduced basis for M rnflllgram(nf , T, M)
determinant of pseudo-matrix M rnfdet(nf , M)
Steinitz class of M rnfsteinitz(nf , M)
ZK-basis of M if ZK-free, or 0 rnfhnfbasis(bnf , M)
n-basis of M, or (n + 1)-generating set rnfbasis(bnf , M)
is M a free ZK-module? rnfisfree(bnf , M)
Graphic Functions
crude graph of expr between a and b plot(X = a, b, expr)
High-resolution plot (immediate plot)
plot expr between a and b ploth(X = a, b, expr, {flag}, {n})
plot points given by lists lx, ly plothraw(lx, ly, {flag})
terminal dimensions plothsizes()
Rectwindow functions
init window w, with size x,y plotinit(w, x, y)
erase window w plotkill(w)
copy w to w2 with offset (dx, dy) plotcopy(w, w2, dx, dy)
clips contents of w plotclip(w)
scale coordinates in w plotscale(w, x1, x2, y1, y2)
ploth in w plotrecth(w, X = a, b, expr, {flag}, {n})
plothraw in w plotrecthraw(w, data, {flag})
draw window w1 at (x1, y1), . . . plotdraw([[w1, x1, y1], . . .])
Low-level Rectwindow Functions
set current drawing color in w to c plotcolor(w, c)
current position of cursor in w plotcursor(w)
write s at cursor’s position plotstring(w, s)
move cursor to (x, y) plotmove(w, x, y)
move cursor to (x + dx, y + dy) plotrmove(w, dx, dy)
draw a box to (x2, y2) plotbox(w, x2, y2)
draw a box to (x + dx, y + dy) plotrbox(w, dx, dy)
draw polygon plotlines(w, lx, ly, {flag})
draw points plotpoints(w, lx, ly)
draw line to (x + dx, y + dy) plotrline(w, dx, dy)
draw point (x + dx, y + dy) plotrpoint(w, dx, dy)
draw point (x + dx, y + dy) plotrpoint(w, dx, dy)
Postscript Functions
as ploth psploth(X = a, b, expr, {flag}, {n})
as plothraw psplothraw(lx, ly, {flag})
as plotdraw psdraw([[w1, x1, y1], . . .])
Based on an earlier version by Joseph H. Silverman
September 2013 v2.27. Copyright c 2013 K. Belabas
Permission is granted to make and distribute copies of this card provided
the copyright and this permission notice are preserved on all copies.
Send comments and corrections to Karim.Belabas@math.u-bordeaux.fr