10.1.1 schemes.hypersurfaces.plane_curves.elliptic.constructor - Elliptic curve constructor

The module schemes.hypersurfaces.plane_curves.elliptic.constructor defines the following methods:

EllipticCurve( x, [y=None])

There are several ways to construct elliptic curves:

- EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given a-invariants. The invariants are coerced into a 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 Cremoa database with the given label. The label is a string, such as"11A" or "37B2".

- 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 j. (Some mild hypothesis on char of base ring.)

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

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

Alternatively, one can create the curve over the 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: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - 1.0000000000000000*x over Complex
Field
sage: E.j_invariant()
2988.9729729729734

sage: E = EllipticCurve(ZZ, [0, 0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Integer Ring

Of course, arithmetic on elliptic curves over Z need not be defined:

sage: P = E([0,0])
sage: P + P + P + P
(2, -3)
sage: P + P + P + P + P
Traceback (most recent call last):
...
ArithmeticError: Point (1/4, -5/8) is not on curve.

EllipticCurve_from_c4c6( c4, c6)

Return an elliptic curve with given $ c_4$ and $ c_6$ 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)

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