Module: sage.schemes.elliptic_curves.constructor
Elliptic curve constructor
Author Log:
Module-level Functions
x, [y=None]) |
There are several ways to construct an elliptic curve:
- EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given a-invariants. The invariants are coerced into the parent of the first element. If all are integers, they are coerced into the rational numbers.
- EllipticCurve([a4,a6]): Same as above, but a1=a2=a3=0.
- EllipticCurve(label): Returns the elliptic curve over Q from the Cremona database with the given label. The label is a string, such as "11a" or "37b2". The letters in the label must be lower case (Cremona's new labeling).
- EllipticCurve(R, [a1,a2,a3,a4,a6]): Create the elliptic curve over R with given a-invariants. Here R can be an arbitrary ring. Note that addition need not be defined.
- EllipticCurve(j): Return an elliptic curve with j-invariant
.
We illustrate creating elliptic curves.
sage: EllipticCurve([0,0,1,-1,0]) Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
We create a curve from a Cremona label:
sage: EllipticCurve('37b2') Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field sage: EllipticCurve('5077a') Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field sage: EllipticCurve('389a') Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
We create curves over a finite field as follows:
sage: EllipticCurve([GF(5)(0),0,1,-1,0]) Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5 sage: EllipticCurve(GF(5), [0, 0,1,-1,0]) Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
The following is a curve over the complex numbers:
sage: E = EllipticCurve(CC, [0,0,1,-1,0]) sage: E Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision sage: E.j_invariant() 2988.97297297297
TESTS:
sage: R = ZZ['u', 'v'] sage: EllipticCurve(R, [1,1]) Elliptic Curve defined by y^2 = x^3 + x +1 over Multivariate Polynomial Ring in u, v over Integer Ring
We create a curve and a point over QQbar:
sage: E = EllipticCurve(QQbar,[0,1]) sage: E(0) (0 : 1 : 0)
c4, c6) |
Return an elliptic curve with given
and
invariants.
sage: E = EllipticCurve_from_c4c6(17, -2005) sage: E Elliptic Curve defined by y^2 = x^3 - 17/48*x + 2005/864 over Rational Field sage: E.c_invariants() (17, -2005)
F, P) |
Given a nonsingular homogenous cubic polynomial F over
in
three variables x, y, z and a projective solution P=[a,b,c] to
F(P)=0, find the minimal Weierstrass equation of the elliptic
curve over
that is isomorphic to the curve defined by
.
Note: USES MAGMA - This function will not work on computers that do not have magma installed. (HELP WANTED - somebody implement this independent of MAGMA.)
First we find that the Fermat cubic is isomorphic to the curve with Cremona label 27a1:
sage: E = EllipticCurve_from_cubic('x^3 + y^3 + z^3', [1,-1,0]) # optional -- requires magma sage: E # optional Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field sage: E.cremona_label() # optional '27a1'
Next we find the minimal model and conductor of the Jacobian of the Selmer curve.
sage: E = EllipticCurve_from_cubic('x^3 + y^3 + 60*z^3', [1,-1,0]) # optional sage: E # optional Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field sage: E.conductor() # optional 24300