Introduction

 
Creation Functions

      Creation of an Elliptic Curve
            EllipticCurve([a, b]) : [ RngElt ] -> CrvEll
            EllipticCurve(f) : RngUPolElt -> CrvEll
            EllipticCurveFromjInvariant(j) : RngElt -> CrvEll
            Example CrvEll_Creation (H131E1)
            EllipticCurve(C) : Sch -> CrvEll, MapSch
            EllipticCurve(C, P) : Crv, Pt -> CrvEll, MapSch
            EllipticCurve(C, pl) : Crv, PlcCrvElt -> CrvEll, MapSch
            SupersingularEllipticCurve(K) : FldFin -> CrvEll
            Example CrvEll_CreationFromCurve (H131E2)
            Example CrvEll_CreationFromCurve2 (H131E3)

      Creation Predicates
            IsEllipticCurve([a, b]) : [ RngElt ] -> BoolElt, CrvEll
            IsEllipticCurve(C) : CrvHyp -> BoolElt, CrvEll, MapIsoSch, MapIsoSch
            Example CrvEll_CreationTest (H131E4)

      Changing the Base Ring
            BaseChange(E, K) : CrvEll, Rng -> CrvEll
            ChangeRing(E, K) : CrvEll, Rng -> CrvEll
            BaseChange(E, h) : CrvEll, Map -> CrvEll
            BaseChange(E, n) : CrvEll, RngIntElt -> CrvEll
            Example CrvEll_BaseExtend (H131E5)

      Alternative Models
            WeierstrassModel(E) : CrvEll -> CrvEll, Map, Map
            IntegralModel(E) : CrvEll -> CrvEll, Map, Map
            SimplifiedModel(E): CrvEll -> CrvEll, Map, Map
            MinimalModel(E) : CrvEll -> CrvEll, Map, Map
            MinimalModel(E, p) : CrvEll, RngIntElt -> CrvEll, Map, Map

      Predicates on Curve Models
            IsWeierstrassModel(E) : CrvEll -> BoolElt
            IsIntegralModel(E) : CrvEll -> BoolElt
            IsSimplifiedModel(E) : CrvEll -> BoolElt
            IsMinimalModel(E) : CrvEll -> BoolElt
            IsIntegralModel(E, P) : CrvEll, RngOrdIdl -> BoolElt
            Example CrvEll_Models (H131E6)

      Twists of Elliptic Curves
            QuadraticTwist(E, d) : CrvEll, RngElt -> CrvEll
            QuadraticTwist(E) : CrvEll -> CrvEll
            QuadraticTwists(E) : CrvEll -> SeqEnum
            Twists(E) : CrvEll -> SeqEnum
            Example CrvEll_QuadraticTwists (H131E7)
            IsTwist(E, F) : CrvEll, CrvEll -> BoolElt
            IsQuadraticTwist(E, F) : CrvEll, CrvEll -> BoolElt, RngElt
            Example CrvEll_NonquadraticTwists (H131E8)
            MinimalQuadraticTwist(E) : CrvEll -> CrvEll, RngIntElt
            Example CrvEll_min_twist (H131E9)

 
Operations on Curves

      Elementary Invariants
            aInvariants(E) : CrvEll -> [ RngElt ]
            bInvariants(E) : CrvEll -> [ RngElt ]
            cInvariants(E) : CrvEll -> [ RngElt ]
            Discriminant(E) : CrvEll -> RngElt
            jInvariant(E) : CrvEll -> RngElt
            HyperellipticPolynomials(E) : CrvEll -> RngUPolElt, RngUPolElt
            Example CrvEll_Invariants (H131E10)
            Example CrvEll_GenericCurve (H131E11)

      Associated Structures
            Category(E) : CrvEll -> Cat
            BaseRing(E) : CrvEll -> Rng

      Predicates on Elliptic Curves
            E eq F : CrvEll, CrvEll -> BoolElt
            E ne F : CrvEll, CrvEll -> BoolElt
            IsIsomorphic(E, F) : CrvEll, CrvEll -> BoolElt, Map
            IsIsogenous(E, F) : CrvEll[FldRat], CrvEll[FldRat] -> BoolElt, Map
            Example CrvEll_Twists2 (H131E12)

 
Polynomials
      DefiningPolynomial(E) : CrvEll -> RngMPolElt
      DivisionPolynomial(E, n) : CrvEll, RngIntElt -> RngUPolElt, RngUPolElt, RngUPolElt
      TwoTorsionPolynomial(E) : CrvEll -> RngMPolElt
      Example CrvEll_DivisionPolynomial (H131E13)

 
Subgroup Schemes

      Creation of Subgroup Schemes
            SubgroupScheme(G, f) : SchGrpEll, RngUPolElt -> SchGrpEll
            TorsionSubgroupScheme(G, n) : SchGrpEll, RngIntElt -> SchGrpEll

      Associated Structures
            Category(G) : SchGrpEll -> Cat
            Curve(G) : SchGrpEll -> CrvEll
            BaseRing(G) : SchGrpEll -> Rng
            DefiningSubschemePolynomial(G) : SchGrpEll -> RngUPolElt

      Predicates on Subgroup Schemes
            G1 eq G2 : SchGrpEll, SchGrpEll -> BoolElt
            G1 ne G2 : SchGrpEll, SchGrpEll -> BoolElt

      Points of Subgroup Schemes
            # G: SchGrpEll -> RngIntElt
            FactoredOrder(G) : SchGrpEll -> RngIntElt
            Points(G) : SchGrpEll -> SetIndx
            Example CrvEll_SubgroupSchemes (H131E14)

 
The Formal Group
      FormalGroupLaw(E, prec) : CrvEll, RngIntElt -> RngMPolElt
      FormalGroupHomomorphism(phi, prec) : MapSch, RngIntElt -> RngSerPowElt
      FormalLog(E) : CrvEll -> RngSerPowElt, PtEll

 
Operations on Point Sets

      Creation of Point Sets
            E(L) : CrvEll, Rng -> SetPtEll
            E(m) : CrvEll, Map -> SetPtEll

      Associated Structures
            Category(H) : SetPtEll -> Cat
            Scheme(H) : SetPtEll -> CrvEll
            Curve(H) : SetPtEll -> CrvEll
            Ring(H) : SetPtEll -> Rng

      Predicates on Point Sets
            H1 eq H2 : SetPtEll, SetPtEll -> BoolElt
            H1 ne H2 : SetPtEll, SetPtEll -> BoolElt
            Example CrvEll_PointSets (H131E15)

 
Morphisms

      Creation Functions
            Example CrvEll_Isogeny (H131E16)
            Isomorphism(E, F, [r, s, t, u]) : CrvEll, CrvEll, SeqEnum -> Map
            Isomorphism(E, F) : CrvEll, CrvEll -> Map
            Automorphism(E, [r, s, t, u]) : CrvEll, SeqEnum -> Map
            IsomorphismData(I) : Map -> [ RngElt ]
            Example CrvEll_Isomorphisms (H131E17)
            IsIsomorphism(I) : Map -> BoolElt, Map
            IsomorphismToIsogeny(I) : Map -> Map
            Example CrvEll_Isomorphism (H131E18)
            TranslationMap(E, P) : CrvEll, PtEll -> Map
            RationalMap(i, t) : Map, Map -> Map
            TwoIsogeny(P) : PtEll -> Map
            Example CrvEll_Map (H131E19)
            IsogenyFromKernel(G) : SchGrpEll -> CrvEll, Map
            IsogenyFromKernelFactored(G) : SchGrpEll -> CrvEll, Map
            IsogenyFromKernel(E, psi) : CrvEll, RngUPolElt -> CrvEll, Map
            IsogenyFromKernelFactored(E, psi) : SchGrpEll, RngUPolElt -> CrvEll, Map
            PushThroughIsogeny(I, v) : Map, RngUPolElt -> RngUPolElt
            DualIsogeny(phi) : Map -> Map
            Example CrvEll_DualIsogeny (H131E20)

      Predicates on Isogenies
            IsZero(I) : Map -> BoolElt
            I eq J : Map, Map -> BoolElt

      Structure Operations
            IsogenyMapPsi(I) : Map -> RngUPolElt
            IsogenyMapPsiMulti(I) : Map -> RngUPolElt
            IsogenyMapPsiSquared(I) : Map -> RngUPolElt
            IsogenyMapPhi(I) : Map -> RngUPolElt
            IsogenyMapPhiMulti(I) : Map -> RngUPolElt
            IsogenyMapOmega(I) : Map -> RngMPolElt
            Kernel(I) : Map -> SchGrpEll
            Degree(I) : Map -> RngIntElt

      Endomorphisms
            MultiplicationByMMap(E, m) : CrvEll, RngIntElt -> Map
            IdentityIsogeny(E) : CrvEll -> Map
            IdentityMap(E) : CrvEll -> Map
            NegationMap(E) : CrvEll -> Map
            FrobeniusMap(E, i) : CrvEll, RngIntElt -> Map
            FrobeniusMap(E) : CrvEll -> Map
            Example CrvEll_Frobenius (H131E21)

      Automorphisms
            AutomorphismGroup(E) : CrvEll -> Grp, Map
            Automorphisms(E) : CrvEll -> SeqEnum

 
Operations on Points

      Creation of Points
            Points(H, x) : SetPtEll, RngElt -> [ PtEll ]
            PointsAtInfinity(H) : SetPtEll -> @ PtEll @

      Creation Predicates
            IsPoint(H, S) : SetPtEll, [ RngElt ] -> BoolElt, PtEll
            IsPoint(H, x) : SetPtEll, RngElt -> BoolElt, PtEll

      Access Operations
            P[i] : PtEll, RngIntElt -> RngElt
            ElementToSequence(P): PtEll -> [ RngElt ]

      Associated Structures
            Category(P) : PtEll -> Cat
            Parent(P) : PtEll -> SetPtEll
            Scheme(P) : SetPtEll -> CrvEll

      Arithmetic
            - P : PtEll -> PtEll
            P + Q : PtEll, PtEll -> PtEll
            P +:= Q : PtEll, PtEll ->
            P - Q : PtEll, PtEll -> PtEll
            P -:= Q : PtEll, PtEll ->
            n * P : RngIntElt, PtEll -> PtEll
            P *:= n : PtEll, RngIntElt ->

      Division Points
            P / n : PtEll, RngIntElt -> PtEll
            P /:= n : PtEll, RngIntElt ->
            DivisionPoints(P, n) : PtEll, RngIntElt -> [ PtEll ]
            IsDivisibleBy(P, n) : PtEll, RngIntElt -> BoolElt, PtEll
            Example CrvEll_PointArithmetic1 (H131E22)
            Example CrvEll_PointArithmetic2 (H131E23)
            Example CrvEll_GenericPoint (H131E24)

      Point Order
            Order(P) : PtEll -> RngIntElt
            FactoredOrder(P) : PtEll -> RngIntElt
            Example CrvEll_PlayWithPoints (H131E25)

      Predicates on Points
            IsId(P) : PtEll -> BoolElt
            P eq Q : PtEll, PtEll -> BoolElt
            P ne Q : PtEll, PtEll -> BoolElt
            P in H : PtEll, SetPtEll -> BoolElt
            P in E : PtEll, CrvEll -> BoolElt
            IsOrder(P, m) : PtEll, RngIntElt -> BoolElt
            IsIntegral(P) : PtEll -> BoolElt
            IsSIntegral(P, S) : PtEll, SeqEnum -> BoolElt
            Example CrvEll_PointPredicates (H131E26)

      Weil Pairing
            WeilPairing(P, Q, n) : PtEll, PtEll, RngIntElt -> RngElt
            IsLinearlyIndependent(S, n) : [ PtEll ], RngIntElt -> BoolElt
            IsLinearlyIndependent(P, Q, n) : PtEll, PtEll, RngIntElt -> BoolElt
            Example CrvEll_WeilPairing (H131E27)

 
Bibliography

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