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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
.
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
or a finite
field.
Next: Elliptic Curves over the
Up: Elliptic Curves [HB 85]
Previous: Elliptic Curves [HB 85]