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A subgroup scheme G of an elliptic curve E is a subscheme
of E defined by a univariate polynomial ψ, and closed under
the group law on E. The points of G are those points of E
whose x-coordinate is a root of ψ. All elliptic curves are
considered to be subgroup schemes with defining polynomial ψ= 0.
Subsections
Creates the subgroup scheme of the subgroup scheme
G defined by the univariate polynomial f.
No checking is done to ensure that the rational points of the result
actually do form a group under the addition law.
Note that G can be an elliptic curve.
Returns the subgroup scheme of n-torsion points of the subgroup scheme
G.
Note that G can be an elliptic curve.
Type(G) : SchGrpEll -> Cat
Returns the category of elliptic curve subgroup schemes, SchGrpEll.
Generic(G) : SchGrpEll -> CrvEll
Returns the elliptic curve E of which G is a subgroup scheme.
CoefficientRing(G) : SchGrpEll -> Rng
The base ring of the subgroup scheme
G; this is the same as the base ring of its curve.
Returns the univariate polynomial that defines G as a subscheme of
its curve.
Returns true if and only if G1 and G2 are subgroup schemes of the
same elliptic curve and are defined by equal polynomials.
The logical negation of eq.
Order(G) : SchGrpEll -> RngIntElt
The order of the group of rational points on the subgroup scheme G.
The factorization of the order of the group of rational points on
the subgroup scheme G.
RationalPoints(G) : SchGrpEll -> SetIndx
The indexed set of rational points of the subgroup scheme G over
its base ring.
We construct a curve over F49 and form several subgroup schemes
from it.
> K<w> := GF(7,2);
> P<t> := PolynomialRing(K);
> E := EllipticCurve([K | 1, 3]);
> G := SubgroupScheme(E, (t-4)*(t-5)*(t-6));
> G;
Subgroup scheme of E defined by x^3 + 6*x^2 + 4*x + 6
> Points(G);
{@ (0 : 1 : 0), (6 : 1 : 1), (6 : 6 : 1), (4 : 1 : 1), (4 : 6 : 1),
(5 : 0 : 1) @}
The points of order 3 form a further subgroup.
> [ Order(P) : P in $1 ];
[ 1, 3, 3, 6, 6, 2 ]
> G2 := SubgroupScheme(G, t - 6);
> G2;
Subgroup scheme of E defined by x + 1
> Points(G2);
{@ (0 : 1 : 0), (6 : 1 : 1), (6 : 6 : 1) @}
We can find this subgroup another way, as the intersection of the
15-torsion points of E and G.
> G3 := TorsionSubgroupScheme(E, 15);
> #G3;
15
> G4 := SubgroupScheme(G3, DefiningSubschemePolynomial(G));
> G4;
Subgroup scheme of E defined by x + 1
> G2 eq G4;
true
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