Elliptic Curves [HB 91]

Removals and Changes:

• Elliptic curves can now be created from general schemes instead of only hyperelliptic curves. Similarly, elliptic curves can now be created from a general scheme and a point rather than a curve and a point. Only one map is returned in each case but it is birational.

• Redesigned routines to put a curve of genus 1 into Weierstrass form. Many commonly occurring cases are recognised and handled more efficiently. The system will automatically choose the most appropriate method.

• Elliptic curves may now be created by specifying the polynomials, similar to the hyperelliptic case. To resolve a naming conflict the former intrinsic EllipticCurve(j) has been changed to EllipticCurveFromjInvariant(j).

New features:

• Elliptic curves can be created from a curve of genus 1 and a place of degree 1.

• HyperellipticPolynomials of an elliptic curve can be retrieved as for hyperelliptic curves.

• Elliptic curves can be created from the corresponding hyperelliptic polynomials.

• The DualIsogeny of an isogeny of an elliptic curve can be obtained as well as the TwoIsogeny of a point of an elliptic curve.