Subgroup Schemes

A subgroup scheme \(G\) of an elliptic curve \(E\) is a subscheme of \(E\) defined by a univariate polynomial \(\psi\) and closed under the group law on \(E\). The points of \(G\) are those points of \(E\) whose \(x\)-coordinate is a root of \(\psi\). All elliptic curves are considered to be subgroup schemes with defining polynomial \(\psi = 0\).

Creation of Subgroup Schemes

SubgroupScheme(G, f): SchGrpEll, RngUPolElt -> SchGrpEll

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.

TorsionSubgroupScheme(G, n): SchGrpEll, RngIntElt -> SchGrpEll

Returns the subgroup scheme of \(n\)-torsion points of the subgroup scheme \(G\). Note that \(G\) can be an elliptic curve.

Associated Structures

Category(G): SchGrpEll -> Cat

Type(G): SchGrpEll -> Cat

Returns the category of elliptic curve subgroup schemes, SchGrpEll.

Curve(G): SchGrpEll -> CrvEll

Generic(G): SchGrpEll -> CrvEll

Returns the elliptic curve \(E\) of which \(G\) is a subgroup scheme.

BaseRing(G): SchGrpEll -> Rng

CoefficientRing(G): SchGrpEll -> Rng

Returns the base ring of the subgroup scheme \(G\); this is the same as the base ring of its curve.

DefiningSubschemePolynomial(G): SchGrpEll -> RngUPolElt

Returns the univariate polynomial that defines \(G\) as a subscheme of its curve.

Predicates on Subgroup Schemes

G1 eq G2: SchGrpEll, SchGrpEll -> BoolElt

Returns true if and only if \(G1\) and \(G2\) are subgroup schemes of the same elliptic curve and are defined by equal polynomials.

G1 ne G2: SchGrpEll, SchGrpEll -> BoolElt

The logical negation of eq.

Points of Subgroup Schemes

# G: SchGrpEll -> RngIntElt

Order(G): SchGrpEll -> RngIntElt

The order of the group of rational points on the subgroup scheme \(G\).

FactoredOrder(G): SchGrpEll -> RngIntElt

The factorisation of the order of the group of rational points on the subgroup scheme \(G\).

Points(G): SchGrpEll -> SetIndx

RationalPoints(G): SchGrpEll -> SetIndx

The indexed set of rational points of the subgroup scheme \(G\) over its base ring.

Example: Subgroup Schemes of a Curve (ex-125594)

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We construct a curve over \({\mathbb{F}}_{49}\) 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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