Elliptic Curves

Elliptic Curves

(PARI-GP version 2.13.0)

An elliptic curve is initially given by 5-tuplev= [a₁, a₂, a₃, a₄, a₆] attached to Weierstrass model or simply [a4, a6]. It must be con- verted to anell struct.

Initializeellstruct over domainD E = ellinit(v,{D= 1})

overQ D= 1

overFp D=p

overFq,q=p^f D=ffgen([p, f])

overQ_p, precisionn D=O(pⁿ)

overC, current bitprecision D= 1.0

over number fieldK D=nf

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

•Edefined overRorC

x-coords. of points of order 2 E.roots

periods / quasi-periods E.omega,E.eta

volume of complex lattice E.area

•Edefined overQ_p

residual characteristic E.p

If|j|p>1: Tate’s [u², u, q,[a, b],L] E.tate

•Edefined overFq

characteristic E.p

#E(Fq)/cyclic structure/generators E.no,E.cyc,E.gen

•Edefined overQ

generators ofE(Q) (requireelldata) E.gen [a1, a2, a3, a4, a6] fromj-invariant ellfromj(j) cubic/quartic/biquadratic to Weierstrass ellfromeqn(eq) add pointsP+Q/P−Q elladd(E, P, Q),ellsub

negate point ellneg(E, P)

computen·P ellmul(E, P, n)

check ifP is onE ellisoncurve(E, P)

order of torsion pointP ellorder(E, P) y-coordinates of point(s) forx ellordinate(E, x) [℘(z), ℘⁰(z)]∈E(C) attached toz∈C ellztopoint(E, z) z∈Csuch thatP= [℘(z), ℘⁰(z)] ellpointtoz(E, P) z∈Q¯^∗/qZtoP∈E( ¯Q_p) ellztopoint(E, z) P∈E( ¯Q_p) toz∈Q¯^∗/qZ ellpointtoz(E, P) Change of Weierstrass models, usingv= [u, r, s, t]

change curveEusingv ellchangecurve(E, v) change pointP usingv ellchangepoint(P, v) change pointP using inverse ofv ellchangepointinv(P, v) Twists and isogenies

quadratic twist elltwist(E, d)

n-division polynomialfn(x) elldivpol(E, n,{x}) [n]P= (φnψn:ωn:ψ_n³); return (φn, ψ_n²) ellxn(E, n,{x})

isogeny fromEtoE/G ellisogeny(E, G)

apply isogeny tog(point or isogeny) ellisogenyapply(f, g) torsion subgroup with generators elltors(E)

Formal group

formal exponential,nterms ellformalexp(E,{n},{x}) formal logarithm,nterms ellformallog(E,{n},{x}) log_E(−x(P)/y(P))∈Q_p;P∈E(Q_p) ellpadiclog(E, p, n, P) P in the formal group ellformalpoint(E,{n},{x}) [ω/dt, xω/dt] ellformaldifferential(E,{n},{x}) w=−1/yin parameter−x/y ellformalw(E,{n},{x})

Curves over finite fields, Pairings

random point onE random(E)

#E(Fq) ellcard(E)

#E(Fq) with almost prime order ellsea(E,{tors}) structureZ/d1Z×Z/d2ZofE(Fq) ellgroup(E)

isEsupersingular? ellissupersingular(E)

Weil pairing ofm-torsion ptsP, Q ellweilpairing(E, P, Q, m) Tate pairing ofP, Q;P m-torsion elltatepairing(E, P, Q, m) Discrete log, findns.t. P= [n]Q elllog(E, P, Q,{ord})

Curves over

Q

Reduction, minimal model

minimal model ofE/Q ellminimalmodel(E,{&v}) quadratic twist of minimal conductor ellminimaltwist(E) [k]P with good reduction ellnonsingularmultiple(E, P) Esupersingular atp? ellissupersingular(E, p) affine points of na¨ıve height≤h ellratpoints(E, h) Complex heights

canonical height ofP ellheight(E, P)

canonical bilinear form taken atP,Q ellheight(E, P, Q) height regulator matrix for pts inL ellheightmatrix(E, L) p-adic heights

cyclotomicp-adic height ofP ∈E(Q) ellpadicheight(E, p, n, P) . . .bilinear form atP, Q∈E(Q) ellpadicheight(E, p, n, P, Q) . . .matrix at vector for pts inLellpadicheightmatrix(E, p, n, L) . . .regulator for canonical height ellpadicregulator(E, p, n, Q) Frobenius onQ_p⊗H_dR¹ (E/Q) ellpadicfrobenius(E, p, n) slope of unit eigenvector of Frobenius ellpadics2(E, p, n) Isogenous curves

matrix of isogeny degrees forQ-isog. curves ellisomat(E) tree of prime degree isogenies ellisotree(E) a modular equation of prime degreeN ellmodulareqn(N) L-function

p-th coeffapofL-function,pprime ellap(E, p) k-th coeffa_kofL-function ellak(E, k) L(E, s) (using less memory thanlfun) elllseries(E, s) L^(r)(E,1) (using less memory thanlfun) ellL1(E, r) a Heegner point onEof rank 1 ellheegner(E) order of vanishing at 1 ellanalyticrank(E,{eps}) root number forL(E, .) atp ellrootno(E,{p}) modular parametrization ofE elltaniyama(E) degree of modular parametrization ellmoddegree(E) compare withH¹(X0(N),Z) (forE⁰→E) ellweilcurve(E) p-adicLfunctionL^(r)_p (E, d, χ^s) ellpadicL(E, p, n,{s},{r},{d}) BSD conjecture forL^(r)_p (E_D, χ⁰) ellpadicbsd(E, p, n,{D= 1}) Iwasawa invariants forLp(E_D, τⁱ) ellpadiclambdamu(E, p, D, i) Elldata package, Cremona’s database:

db code"11a1"↔[conductor,class,index] ellconvertname(s) generators of Mordell-Weil group ellgenerators(E)

look upEin database ellidentify(E)

all curves matching criterion ellsearch(N) loop over curves with cond. fromatob forell(E, a, b,seq)

Curves over number field

K

coeffapofL-function ellap(E,

p

)

Kodaira type of

p

-fiber ofE elllocalred(E,

p

) integral model ofE/K ellintegralmodel(E,{&v}) minimal model ofE/K ellminimalmodel(E,{&v}) minimal discriminant ofE/K ellminimaldisc(E) cond, min mod, Tamagawa num[N, v, c] ellglobalred(E)

global Tamagawa number elltamagawa(E)

P∈E(K)n-divisible? [n]Q=P ellisdivisible(E, P, n,{&Q}) L-function

A domainD= [c, w, h] in initialization mean we restricts∈Cto domain|<(s)−c|< w,|=(s)|< h;D= [w, h] encodes [1/2, w, h]

and [h] encodesD= [1/2,0, h] (critical line up to heighth).

vector of firstn a_k’s inL-function ellan(E, n)

initL^(k)(E, s) fork≤n L = lfuninit(E, D,{n= 0}) computeL(E, s) (n-th derivative) lfun(L, s,{n= 0}) L(E,1, r)/(r!·R·#Sha) assuming BSD ellbsd(E)

Other curves of small genus

A hyperelliptic curve is given by a pair [P, Q] (y²+Qy=P with Q²+ 4P squarefree) or a single squarefree polynomialP(y²=P).

reduction ofy²+Qy=P (genus 2) genus2red([P, Q],{p}) affine rational points of height≤h hyperellratpoints([P, Q], h) find a rational point on a conic,^txGx= 0 qfsolve(G)

quadratic Hilbert symbol (atp) hilbert(x, y,{p}) all solutions inQ³of ternary form qfparam(G, x) P, Q∈Fq[X]; char. poly. of Frobenius hyperellcharpoly([P, Q]) matrix of Frobenius onQ_p⊗H_dR¹ hyperellpadicfrobenius

Elliptic & Modular Functions

w= [ω1, ω2] orellstruct (E.omega),τ=ω1/ω2. arithmetic-geometric mean agm(x, y) ellipticj-function 1/q+ 744 +· · · ellj(x)

Weierstrassσ/℘/ζfunction ellsigma(w, z),ellwp,ellzeta periods/quasi-periods ellperiods(E,{flag}),elleta(w) (2iπ/ω2)^kE_k(τ) elleisnum(w, k,{flag}) modified Dedekindηfunc. Q

(1−qⁿ) eta(x,{flag})

Dedekind sums(h, k) sumdedekind(h, k)

Jacobi sine theta function theta(q, z) k-th derivative at z=0 oftheta(q, z) thetanullk(q, k)

Weber’sf functions weber(x,{flag})

modular pol. of levelN polmodular(N,{inv = j}) Hilbert class polynomial forQ(√

D) polclass(D,{inv = j})

Based on an earlier version by Joseph H. Silverman October 2020 v2.37. Copyright c2020 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[email protected]