General Elliptic Curves

Removals and Changes:

• The category of elliptic curves CurveEll and their points CurveEllPt has been changed to CrvEll and CrvEllPt.
• The function TwoTorsionPolynomial has been changed to return the 2-torsion polynomial as a multivariate polynomial. The syntax DivisionPolynomial(E, 2) continues to return the univariate polynomial as the first return value.
• The function TwoTorsionPolynomialMultivariate is now deprecated and will be removed in a future release.
• The function IsOrderOfPoint has been changed to IsOrder; the previous syntax is now deprecated and will be removed in a future release.
• The syntax BaseChange(E, K, j), where j is a homomorphism from BaseRing(E) to K has been changed to BaseChange(E,j); the previous syntax is now deprecated and will be removed in a future release. Similarly for BaseExtend.
• The syntax EllipticCurve(K,j), where j is the j-invariant of a curve over K, has been changed to EllipticCurve(j); the previous syntax is now deprecated and will be removed in a future release.

New features:

• Elliptic curves over any field can be now created with given j-invariant.
• Elliptic curves can now be created from a plane curve C of genus one with a give rational point P (function EllipticCurve(C,P)). The transformation from a hyperelliptic curve of genus one, having a unique point at infinity, is achieved with EllipticCurve(C).
• New function DivisionFunction(E) to return the division polynomial as an element of the affine coordinate ring of E.