Elliptic Curves

Little has been implemented so far.

ModularForm(E) : CrvEll -> ModFrmElt

Newform(E) : CrvEll -> ModFrmElt

The modular form associated to the elliptic curve E (which must be defined over the rationals).

Eigenform(E, prec) : CrvEll, RngIntElt -> RngSerPowElt

qEigenform(E, prec) : CrvEll, RngIntElt -> RngSerPowElt

The q-expansion of the newform associated to E, to the specified precision.

Note: this is exactly the same as calling qExpansion(ModularForm(E),prec).

EllipticCurve(f) : ModFrmElt -> CrvEll

An elliptic curve E with associated modular form f, when f is a weight 2 newform on Γ₀(N) with rational Fourier coefficients.

The Cremona database is used to identify the isogeny class. (A routine to compute the curve from scratch is implemented, and can be called with EllipticCurve(M : Database:=false) where M is the relevant space of modular symbols; however this is not optimized for large level.)

> M := ModularForms(Gamma0(389),2);
> f := Newform(M,1);
> Degree(f);
1
> E := EllipticCurve(f);
> E;
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational 
Field
> Conductor(E);
389
> time s := PowerSeries(f,200);  // faster because it knows the elliptic curve
Time: 0.509