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
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.weierstrass_model() Elliptic Curve defined by y^2 = x^3 - 13392*x - 1080432 over Rational Field sage: E.discriminant() -161051
Given an elliptic curve defined over
,
SAGE can compute its set or
-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 : 1 : 0), (0 : 0 : 1), (0 : 4 : 1), (1 : 0 : 1), (1 : 4 : 1)] sage: E.cardinality() 5 sage: G = E.abelian_group(); G (Multiplicative Abelian Group isomorphic to C5, ((1 : 0 : 1),)) sage: G[0].permutation_group() Permutation Group with generators [(1,2,3,4,5)]
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