Let
be a ``nice'' elliptic curve whose equation has integer coefficients,
let
be the conductor of
and, for each
, let
be the number appearing in the Hasse-Weil
-function of
.
The Taniyama-Shimura conjecture (proven by Wiles) states that there
exists a modular form of weight two and level
which is an
eigenform under the Hecke operators and has a Fourier series
. Sage can compute the sequence
associated to
. Here is an example.
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field sage: E.conductor() 11 sage: E.anlist(20) [0, 1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2] sage: E.analytic_rank() 0
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