Elliptic Curves
- Introduction
- Creation Functions
- Creation of an Elliptic Curve
EllipticCurve([a, b]): [ RngElt ] → CrvEllEllipticCurve([a1, a2, a3, a4, a6]): [ RngElt ] → CrvEllEllipticCurve(f): RngUPolElt → CrvEllEllipticCurve(f, h): RngUPolElt, RngUPolElt → CrvEllEllipticCurveFromjInvariant(j): RngElt → CrvEllEllipticCurveWithjInvariant(j): RngElt → CrvEllExample: CreationEllipticCurve(C): Sch → CrvEll, MapSchEllipticCurve(C, P): Crv, Pt → CrvEll, MapSchEllipticCurve(C, pl): Crv, PlcCrvElt → CrvEll, MapSchSupersingularEllipticCurve(K): FldFin → CrvEllExample: Creation From CurveExample: Creation From Curve2
- Creation Predicates
IsEllipticCurve([a, b]): [ RngElt ] → BoolElt, CrvEllIsEllipticCurve([a1, a2, a3, a4, a6]): [ RngElt ] → BoolElt, CrvEllIsEllipticCurve(C): CrvHyp → BoolElt, CrvEll, MapIsoSch, MapIsoSchExample: Creation Test
- Changing the Base Ring
BaseChange(E, K): CrvEll, Rng → CrvEllBaseExtend(E, K): CrvEll, Rng → CrvEllChangeRing(E, K): CrvEll, Rng → CrvEllBaseChange(E, h): CrvEll, Map → CrvEllBaseExtend(E, h): CrvEll, Map → CrvEllBaseChange(E, n): CrvEll, RngIntElt → CrvEllBaseExtend(E, n): CrvEll, RngIntElt → CrvEllExample: Base Extend
- Alternative Models
WeierstrassModel(E): CrvEll → CrvEll, Map, MapIntegralModel(E): CrvEll → CrvEll, Map, MapSimplifiedModel(E): CrvEll → CrvEll, Map, MapMinimalModel(E): CrvEll → CrvEll, Map, MapMinimalModel(E, p): CrvEll, RngIntElt → CrvEll, Map, MapMinimalModel(E, P : parameters): CrvEll, RngOrdIdl → CrvEll, Map
- Predicates on Curve Models
- Twists of Elliptic Curves
- Operations on Curves
- Elementary Invariants
- Associated Structures
- Predicates on Elliptic Curves
E eq F: CrvEll, CrvEll → BoolEltE ne F: CrvEll, CrvEll → BoolEltIsIsomorphic(E, F): CrvEll, CrvEll → BoolElt, MapIsIsogenous(E, F): CrvEll[FldRat], CrvEll[FldRat] → BoolElt, MapIsIsogenous(E, F): CrvEll[FldFin], CrvEll[FldFin] → BoolEltExample: Twists2
- Polynomials
DefiningPolynomial(E): CrvEll → RngMPolEltDivisionPolynomial(E, n): CrvEll, RngIntElt → RngUPolElt, RngUPolElt, RngUPolEltDivisionPolynomial(E, n, g): CrvEll, RngIntElt, RngUPolElt → RngUPolElt, RngUPolElt, RngUPolEltTwoTorsionPolynomial(E): CrvEll → RngMPolEltExample: Division Polynomial- Complex Multiplication Division Polynomials
- Subgroup Schemes
- The Formal Group
- Operations on Point Sets
- Morphisms
- Creation Functions
Example: IsogenyIsomorphism(E, F, [r, s, t, u]): CrvEll, CrvEll, SeqEnum → MapIsomorphism(E, F): CrvEll, CrvEll → MapAutomorphism(E, [r, s, t, u]): CrvEll, SeqEnum → MapIsomorphismData(I): Map → [ RngElt ]Example: IsomorphismsIsIsomorphism(I): Map → BoolElt, MapIsomorphismToIsogeny(I): Map → MapExample: IsomorphismTranslationMap(E, P): CrvEll, PtEll → MapRationalMap(i, t): Map, Map → MapTwoIsogeny(P): PtEll → MapExample: MapIsogenyFromKernel(G): SchGrpEll → CrvEll, MapIsogenyFromKernelFactored(G): SchGrpEll → CrvEll, MapIsogenyFromKernel(E, psi): CrvEll, RngUPolElt → CrvEll, MapIsogenyFromKernelFactored(E, psi): SchGrpEll, RngUPolElt → CrvEll, MapPushThroughIsogeny(I, v): Map, RngUPolElt → RngUPolEltPushThroughIsogeny(I, G): Map, SchGrpEll → SchGrpEllDualIsogeny(phi): Map → MapExample: Dual Isogeny
- Predicates on Isogenies
- Structure Operations
- Endomorphisms
- Automorphisms
- Operations on Points
- Creation of Points
H ! [x, y, z]: SetPtEll, [ RngElt ] → PtEllelt< H | x, y, z >: SetPtEll, RngElt, RngElt, RngElt → PtEllE ! [x, y, z]: CrvEll, [ RngElt ] → PtEllelt< E | x, y, z >: CrvEll, RngElt, RngElt, RngElt → PtEllH ! 0: SetPtEll, RngIntElt → PtEllId(H): SetPtEll → PtEllIdentity(H): SetPtEll → PtEllE ! 0: CrvEll, RngIntElt → PtEllId(E): CrvEll → PtEllIdentity(E): CrvEll → PtEllPoints(H, x): SetPtEll, RngElt → [ PtEll ]Points(E, x): CrvEll, RngElt → [ PtEll ]Points(H): SetPtEll → {@ PtEll @}RationalPoints(H): SetPtEll → {@ PtEll @}Points(E): CrvEll → {@ PtEll @}RationalPoints(E): CrvEll → {@ PtEll @}PointsAtInfinity(H): SetPtEll → {@ PtEll @}PointsAtInfinity(E): CrvEll → {@ PtEll @}
- Creation Predicates
IsPoint(H, S): SetPtEll, [ RngElt ] → BoolElt, PtEllIsPoint(E, S): CrvEll, [ RngElt ] → BoolElt, PtEllIsPoint(H, x): SetPtEll, RngElt → BoolElt, PtEllIsPoint(E, x): CrvEll, RngElt → BoolElt, PtEll
- Access Operations
- Associated Structures
- Arithmetic
- Division Points
- Point Order
- Predicates on Points
- Weil Pairing
WeilPairing(P, Q, n): PtEll, PtEll, RngIntElt → RngEltIsLinearlyIndependent(S, n): [ PtEll ], RngIntElt → BoolEltIsLinearlyIndependent(P, Q, n): PtEll, PtEll, RngIntElt → BoolEltExample: Weil Pairing
