General Elliptic Curves

Removals and Changes:

• Elliptic curves are now scheme types (of type CrvEll still), and inherit all of the appropriate scheme functionality. As part of this change, the curve is no longer the parent of points but various point sets are instead. The type of a point is now PtEll and the type of a point set is SetPtEll.
• Subgroup schemes of elliptic curves are now also schemes, of type SchGrpEll. All functions which apply to subgroup schemes should also apply to the elliptic curves.
• Isomorphisms are not compatible with isogenies at the moment.
• In keeping with the scheme philosophy, BaseExtend(E, K) does not work unless K is an extension of the base field of E. In particular, this will fail if K is a finite field and E is defined over $\mbox{\bf Q}$. The intrinsic ChangeRing can be used instead in this case.
• Subschemes of elliptic curves are now constructed using scheme functions, and are no longer a special type.
• The deprecated function EllipticCurve(K, j) has been removed; use EllipticCurve(K!j) instead.
• The deprecated function IsPoint(S, E) has been removed; use IsPoint(E, S) instead.
• The deprecated function IsOrderOfPoint has been removed; use IsOrder instead.
• The functions mTorsionSubgroup, nTorsionSubgroup, and pTorsionSubgroup have been removed; use TorsionSubgroupScheme instead.
• The functions Lift(E, K) and Lift(E, K, m) have been removed; use BaseExtend or ChangeRing instead.
• The function Subgroup has been renamed to SubgroupScheme.
• Since elliptic curves are also (trivial) subgroup schemes, the function DefiningPolynomial for a subgroup scheme has been renamed to DefiningSubschemePolynomial to make it clear which function is meant.
• The functions WeierstrassCoefficients(E) and RationalPoints(E, K) should be considered deprecated and will be removed in a future release.

New features:

• Isomorphisms between elliptic curves are now scheme maps, and inherit the appropriate functionality of these maps.
• The function PointsAtInfinity has been added for symmetry with the hyperelliptic curves.
• Elliptic curves can be created from general curves which are already in the correct form, without the need to specify a point on the curve.
• The functions IsIsomorphic and SimplifiedModel now work for curves defined over any field which supports root finding, not just finite fields.
• The function Isomorphism has been added to return the isomorphism between two curves that are known to be isomorphic.
• The functions IsogenyFromKernel and IsogenyFromKernelFactored are no longer restricted to elliptic curves defined over $\mbox{\bf Q}$ or a finite field.