13.4 Modular form associated to an elliptic curve over $ \mathbb{Q}$

Let $ E$ be a ``nice'' elliptic curve whose equation has integer coefficients, let $ N$ be the conductor of $ E$ and, for each $ n$ , let $ a_n$ be the number appearing in the Hasse-Weil $ L$ -function of $ E$ . The Taniyama-Shimura conjecture (proven by Wiles) states that there exists a modular form of weight two and level $ N$ which is an eigenform under the Hecke operators and has a Fourier series $ \sum_{n = 0}^\infty a_n q^n$ . Sage can compute the sequence $ a_n$ associated to $ E$ . 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

See About this document... for information on suggesting changes.