.. index:: elliptic curves *************** Elliptic curves *************** Conductor ========= How do you compute the conductor of an elliptic curve (over :math:\QQ) in Sage? Once you define an elliptic curve :math:E in Sage, using the EllipticCurve command, the conductor is one of several "methods" associated to :math:E. Here is an example of the syntax (borrowed from section 2.4 "Modular forms" in the tutorial): :: sage: E = EllipticCurve([1,2,3,4,5]) sage: E Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field sage: E.conductor() 10351 :math:j-invariant ===================== How do you compute the :math:j-invariant of an elliptic curve in Sage? Other methods associated to the EllipticCurve class are j_invariant, discriminant, and weierstrass_model. Here is an example of their syntax. :: 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.j_invariant() -122023936/161051 sage: E.short_weierstrass_model() Elliptic Curve defined by y^2 = x^3 - 13392*x - 1080432 over Rational Field sage: E.discriminant() -161051 sage: E = EllipticCurve(GF(5),[0, -1, 1, -10, -20]) sage: E.short_weierstrass_model() Elliptic Curve defined by y^2 = x^3 + 3*x + 3 over Finite Field of size 5 sage: E.j_invariant() 4 .. index:: elliptic curves The :math:GF(q)-rational points on E ======================================== How do you compute the number of points of an elliptic curve over a finite field? Given an elliptic curve defined over :math:\mathbb{F} = GF(q), Sage can compute its set of :math:\mathbb{F}-rational points :: sage: E = EllipticCurve(GF(5),[0, -1, 1, -10, -20]) sage: E Elliptic Curve defined by y^2 + y = x^3 + 4*x^2 over Finite Field of size 5 sage: E.points() [(0 : 0 : 1), (0 : 1 : 0), (0 : 4 : 1), (1 : 0 : 1), (1 : 4 : 1)] sage: E.cardinality() 5 sage: G = E.abelian_group() sage: G Additive abelian group isomorphic to Z/5 embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 4*x^2 over Finite Field of size 5 sage: G.permutation_group() Permutation Group with generators [(1,2,3,4,5)] .. index:: pair: modular form; elliptic curve Modular form associated to an elliptic curve over :math:\QQ ======================================================================== Let :math:E be a "nice" elliptic curve whose equation has integer coefficients, let :math:N be the conductor of :math:E and, for each :math:n, let :math:a_n be the number appearing in the Hasse-Weil :math:L-function of :math:E. The Taniyama-Shimura conjecture (proven by Wiles) states that there exists a modular form of weight two and level :math:N which is an eigenform under the Hecke operators and has a Fourier series :math:\sum_{n = 0}^\infty a_n q^n. Sage can compute the sequence :math:a_n associated to :math: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