Introduction

 
Curves over the Rationals

      Local Invariants
            Conductor(E) : CrvEll -> RngIntElt
            BadPrimes(E) : CrvEll -> [ RngIntElt ]
            TamagawaNumber(E, p) : CrvEll, RngIntElt -> RngIntElt
            TamagawaNumbers(E) : CrvEll -> [ RngIntElt ]
            LocalInformation(E, p) : CrvEll, RngIntElt -> <RngIntElt, RngIntElt, RngIntElt, RngIntElt, SymKod, BoolElt>, CrvEll
            LocalInformation(E) : CrvEll -> [ Tup ]
            ReductionType(E, p) : CrvEll, RngIntElt -> MonStgElt
            TraceOfFrobeniusDirect(E, p) : CrvEll, RngIntElt -> RngIntElt
            TracesOfFrobenius(E, B) : CrvEll, RngIntElt -> SeqEnum
            Example CrvEllQNF_frobenius-traces (H133E1)

      Kodaira Symbols
            KodairaSymbol(E, p) : CrvEll, RngIntElt -> SymKod
            KodairaSymbols(E) : CrvEll -> [ SymKod ]
            KodairaSymbol(s) : MonStgElt -> SymKod
            h eq k : SymKod, SymKod -> BoolElt
            h ne k : SymKod, SymKod -> BoolElt
            Example CrvEllQNF_Kodaira (H133E2)

      Complex Multiplication
            HasComplexMultiplication(E) : CrvEll -> BoolElt, RngIntElt

      Isogenous Curves
            IsogenousCurves(E) : CrvEll[FldRat] -> SeqEnum, RngIntElt
            FaltingsHeight(E) : CrvEll[FldRat] -> FldReElt
            StableFaltingsHeight(E) : CrvEll[FldRat] -> FldReElt
            Example CrvEllQNF_isog-curves (H133E3)

      Heights and Height Pairing
            NaiveHeight(P) : PtEll -> FldPrElt
            Height(P: parameters) : PtEll -> NFldComElt
            LocalHeight(P, p) : PtEll, RngIntElt -> FldComElt
            HeightPairing(P, Q: parameters) : PtEll, PtEll -> FldComElt
            HeightPairingMatrix(S: parameters) : [PtEll] -> AlgMat
            Regulator(S) : [ PtEll ] -> FldComElt
            Regulator(E) : CrvEll -> FldComElt
            Example CrvEllQNF_FunWithHeights (H133E4)
            SilvermanBound(H) : SetPtEll -> FldPrElt
            SiksekBound(H: parameters) : SetPtEll -> FldPrElt
            Example CrvEllQNF_Bounds (H133E5)
            IsLinearlyIndependent(P, Q) : PtEll, PtEll -> BoolElt, ModTupElt
            IsLinearlyIndependent(S) : [ PtEll ] -> BoolElt, ModTupElt
            ReducedBasis(S) : [ PtEll ] -> [ PtEll ]
            Example CrvEllQNF_LinearIndependence (H133E6)
            pAdicHeight(P, p) : PtEll, RngIntElt -> FldPadElt
            pAdicRegulator(S, p) : [PtEll], RngIntElt -> FldPadElt
            EisensteinTwo(E, p) : CrvEll, RngIntElt -> FldPadElt
            FrobeniusMatrix(E, p) : CrvEll, RngIntElt -> Mtrx
            Example CrvEllQNF_padic-height (H133E7)

      Heegner Points
            HeegnerPoint(E : parameters) : CrvEll -> BoolElt, PtEll
            HeegnerPoint(C : parameters) : CrvHyp -> BoolElt, PtHyp
            ModularParametrization(E, z, B : parameters) : CrvEll[FldRat], FldComElt, RngIntElt -> FldComElt
            HeegnerDiscriminants(E,lo,hi) : CrvEll[FldRat], RngIntElt, RngIntElt -> SeqEnum
            HeegnerForms(E,D : parameters) : CrvEll[FldRat], RngIntElt -> SeqEnum
            HeegnerForms(N,D : parameters) : RngIntElt, RngIntElt -> SeqEnum
            ManinConstant(E) : CrvEll[FldRat] -> RngIntElt
            HeegnerTorsionElement(E, Q) : CrvEll[FldRat], RngIntElt -> PtEll
            HeegnerPoints(E, D : parameters) : CrvEll[FldRat], RngIntElt -> Tup, PtEll
            Example CrvEllQNF_Heegner (H133E8)
            Example CrvEllQNF_Heegner2 (H133E9)
            Example CrvEllQNF_Heegner3 (H133E10)
            Example CrvEllQNF_Heegner4 (H133E11)
            Example CrvEllQNF_Heegner5 (H133E12)

      Analytic Information
            Periods(E: parameters) : CrvEll -> [ FldComElt ]
            Periods(E, k) : CrvEll, RngIntElt -> [ FldComElt ]
            EllipticCurveFromPeriods(om: parameters) : [ FldComElt ] -> CrvEll
            RealPeriod(E: parameters) : CrvEll -> FldReElt
            EllipticExponential(E, z) : CrvEll, FldComElt -> [ FldComElt ]
            EllipticExponential(E, k, z) : CrvEll, RngIntElt, FldComElt -> [ FldComElt ]
            EllipticExponential(E, S) : CrvEll, [ FldRat ] -> [ FldComElt ]
            EllipticLogarithm(P): PtEll[FldRat] -> FldComElt
            EllipticLogarithm(P, k): PtEll[FldNum], RngIntElt -> FldComElt
            EllipticLogarithm(E, S): CrvEll, [ FldComElt ] -> FldComElt
            pAdicEllipticLogarithm(P, p: parameters): PtEll, RngIntElt -> FldLocElt
            Example CrvEllQNF_ell-exp (H133E13)
            Example CrvEllQNF_ellexp-nf (H133E14)
            RootNumber(E) : CrvEll -> RngIntElt
            RootNumber(E, p) : CrvEll, RngIntElt -> RngIntElt
            AnalyticRank(E) : CrvEll -> RngIntElt, FldReElt
            ConjecturalRegulator(E) : CrvEll -> FldReElt, RngIntElt
            ConjecturalRegulator(E, v) : CrvEll, FldReElt -> FldReElt
            Example CrvEllQNF_analytic-rank (H133E15)
            Example CrvEllQNF_conjectural-regulator (H133E16)
            ModularDegree(E) : CrvEll -> RngIntElt
            Example CrvEllQNF_mod-deg (H133E17)

      Integral and S-integral Points
            IntegralPoints(E) : CrvEll[FldRat] -> [ PtEll ]
            SIntegralPoints(E, S) : CrvEll, SeqEnum -> [ PtEll ]
            Example CrvEllQNF_IntegralPoints (H133E18)
            Example CrvEllQNF_SIntegralPoints (H133E19)
            IntegralQuarticPoints(Q) : [ RngIntElt ] -> [ SeqEnum ]
            IntegralQuarticPoints(Q, P) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
            SIntegralQuarticPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
            Example CrvEllQNF_IntegralPointsSequence (H133E20)
            SIntegralLjunggrenPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
            SIntegralDesbovesPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
            Example CrvEllQNF_Desboves (H133E21)

      Elliptic Curve Database
            EllipticCurveDatabase(: parameters) : -> DB
            SetBufferSize(D, n) : DB, RngIntElt ->
            LargestConductor(D) : DB -> RngIntElt
            ConductorRange(D) : DB -> RngIntElt, RngIntElt
            # D : DB -> RngIntElt
            NumberOfCurves(D, N) : DB, RngIntElt -> RngIntElt
            NumberOfCurves(D, N, i) : DB, RngIntElt, RngIntElt -> RngIntElt
            NumberOfIsogenyClasses(D, N) : DB, RngIntElt -> RngIntElt
            EllipticCurve(D, N, I, J): DB, RngIntElt, RngIntElt, RngIntElt -> CrvEll
            EllipticCurve(D, S): DB, MonStgElt -> CrvEll
            Random(D) : DB -> CrvEll
            CremonaReference(D, E) : DB, CrvEll -> MonStgElt
            Example CrvEllQNF_ecdb1 (H133E22)
            EllipticCurves(D, N, I) : DB, RngIntElt, RngIntElt -> [ CrvEll ]
            EllipticCurves(D, N) : DB, RngIntElt -> [ CrvEll ]
            EllipticCurves(D, S) : DB, MonStgElt -> [ CrvEll ]
            EllipticCurves(D) : DB -> [ CrvEll ]
            Example CrvEllQNF_ecdb2 (H133E23)

 
Curves over Number Fields

      Local Invariants
            Conductor(E) : CrvEll -> RngOrdIdl
            BadPlaces(E) : CrvEll -> SeqEnum
            BadPlaces(E, L) : CrvEll, FldNum -> SeqEnum
            LocalInformation(E, P) : CrvEll, RngOrdIdl -> Tup, CrvEll
            LocalInformation(E) : CrvEll -> [ Tup ]
            Reduction(E, p) : CrvEll, RngOrdIdl -> CrvEll, Map

      Complex Multiplication
            HasComplexMultiplication(E) : CrvEll -> BoolElt, RngIntElt

      Heights
            NaiveHeight(P) : PtEll -> FldPrElt
            Height(P : parameters) : PtEll -> FldPrElt
            HeightPairingMatrix(P : parameters) : [PtEll] -> AlgMatElt
            LocalHeight(P, Pl : parameters) : PtEll, PlcNumElt -> FldPrElt
            HeightDifferenceBounds(E) : CrvEll -> FldReElt, FldReElt
            CPSHeightBounds(E) : CrvEll -> FldReElt, FldReElt
            SilvermanHeightBounds(E) : CrvEll -> FldReElt, FldReElt

      Integral Points
            IntegralPoints(E) : CrvEll[FldNum] -> [ PtEll ]

      Elliptic Curve Chabauty
            Chabauty(MWmap, Ecov) : Map, MapSch -> SetEnum, RngIntElt
            Chabauty(MWmap, Ecov, p) : Map, MapSch, RngIntElt -> RngIntElt, SetEnum, RngIntElt, Tup
            Example CrvEllQNF_ECchabauty (H133E24)

      Auxiliary Functions for Etale Algebras
            AbsoluteAlgebra(A) : RngUPolRes -> SetCart, Map
            pSelmerGroup(A, p, S) : RngUPolRes, RngIntElt, SetEnum[RngOrdIdl] -> GrpAb, Map
            LocalTwoSelmerMap(P) : RngOrdIdl -> Map
            LocalTwoSelmerMap(A, P) : RngUPolRes, RngOrdIdl -> Map, SeqEnum
            Example CrvEllQNF_selmer-etale (H133E25)

      Analytic Information
            RootNumber(E, P) : CrvEll, RngOrdIdl -> RngIntElt
            RootNumber(E) : CrvEll -> RngIntElt
            AnalyticRank(E) : CrvEll -> RngIntElt, FldReElt
            ConjecturalRegulator(E) : CrvEll -> FldReElt, RngIntElt
            ConjecturalSha(E, Pts) : CrvEll, SeqEnum[PtEll] -> FldReElt

      Elliptic Curves of Given Conductor
            EllipticCurveSearch(N, Effort) : RngOrdIdl, RngIntElt -> SeqEnum

 
Curves over p-adic Fields

      Local Invariants
            Conductor(E) : CrvEll -> FldPadElt
            LocalInformation(E) : CrvEll -> Tup, CrvEll
            RootNumber(E) : CrvEll -> RngIntElt

 
Mordell--Weil Groups and Descent Methods

      Torsion
            TorsionSubgroup(E) : CrvEll -> GrpAb, Map
            TwoTorsionSubgroup(E) : CrvEll -> GrpAb, Map
            TorsionBound(E, n) : CrvEll, RngIntElt -> RngIntElt
            pPowerTorsion(E, p) : CrvEll, RngIntElt -> GrpAb, Map

      Mordell--Weil Group and Rank
            RankBounds(H: parameters) : SetPtEll -> RngIntElt, RngIntElt
            Rank(H: parameters) : SetPtEll -> RngIntElt, BoolElt
            MordellWeilGroup(H: parameters) : SetPtEll -> GrpAb, Map, BoolElt, BoolElt
            Generators(H) : SetPtEll -> [ PtEll ]
            NumberOfGenerators(H) : SetPtEll -> RngIntElt
            Saturation(points, n) : [ PtEll ], RngIntElt -> [ PtEll ]
            Saturation(points) : [ PtEll ] -> [ PtEll ]
            Example CrvEllQNF_MordellWeil (H133E26)
            Example CrvEllQNF_Rank (H133E27)
            MordellWeilShaInformation(E: parameters) : CrvEll -> [RngIntElt], [PtEll], [Tup]
            Example CrvEllQNF_mwsha-example (H133E28)

      Two-Descent
            TwoDescent(E: parameters) : CrvEll -> [CrvHyp] , [Map], Map
            AssociatedEllipticCurve(f) : RngUPolElt -> CrvEll, Map
            TwoCover(e) : FldNumElt -> CrvHyp, Map
            Example CrvEllQNF_twodescent (H133E29)

            Two Descent Using Isogenies
                  TwoIsogenyDescent(E : parameters) : CrvEll -> SeqEnum[CrvHyp], List, SeqEnum[CrvHyp], List, MapSch, MapSch
                  LiftDescendant(C) : CrvHyp -> SeqEnum[ CrvHyp ], List, MapSch

            Invariants
                  QuarticIInvariant(q) : RngUPolElt -> RngIntElt
                  QuarticNumberOfRealRoots(q) : RngUPolElt -> RngUPolElt
                  QuarticMinimise(q) : RngUPolElt -> RngUPolElt, AlgMatElt
                  QuarticReduce(q) : RngUPolElt -> RngUPolElt, AlgMatElt
                  IsEquivalent(f,g) : RngUPolElt, RngUPolElt -> BoolElt

      Selmer Groups
            DescentMaps(phi) : Map -> Map, Map
            SelmerGroup(phi) : Map -> GrpAb, Map, Map, SeqEnum, SetEnum
            TwoSelmerGroup(E) : CrvEll -> GrpAb, Map, SetEnum, Map, SeqEnum
            Example CrvEllQNF_selmer (H133E30)
            Example CrvEllQNF_selmer2 (H133E31)
            Example CrvEllQNF_selmer3 (H133E32)
            Example CrvEllQNF_selmer4 (H133E33)

      The Cassels-Tate Pairing
            CasselsTatePairing(C, D) : CrvHyp, CrvHyp -> RngIntElt
            CasselsTatePairing(C, D) : Crv, CrvHyp -> RngIntElt
            Example CrvEllQNF_cassels-tate-example (H133E34)

      Four-Descent
            FourDescent(C : parameters) : CrvHyp -> [Crv]
            Example CrvEllQNF_simplefourdesc (H133E35)
            AssociatedEllipticCurve(qi) : Crv -> CrvEll, Map
            QuadricIntersection(F) : [AlgMatElt] -> Crv
            QuadricIntersection(E) : CrvEll -> Crv, MapIsoSch
            IsQuadricIntersection(C) : Crv -> BoolElt, [AlgMatElt]
            PointsQI(C, B : parameters) : Crv, RngIntElt -> [Pt]
            TwoCoverPullback(H, pt) : CrvHyp[FldRat], PtEll[FldRat] -> [PtHyp]
            FourCoverPullback(C, pt) : Crv[FldRat], PtEll[FldRat] -> [Pt]
            Example CrvEllQNF_fourdescent (H133E36)

      Eight-Descent
            EightDescent(C : parameters) : Crv -> [ Crv ], [ MapSch ]

      Three-Descent and Five-Descent
            ThreeDescent(E : parameters) : CrvEll -> [ Crv ], List
            Example CrvEllQNF_selmer-famous-example (H133E37)
            ThreeSelmerGroup(E : parameters) : CrvEll -> GrpAb, Map
            ThreeDescentCubic(E, α : parameters) : CrvEll, Tup -> Crv, MapSch
            ThreeIsogenyDescent(E : parameters) : CrvEll -> [ Crv ], List, [ Crv ], List, MapSch
            ThreeIsogenySelmerGroups(E : parameters) : CrvEll -> GrpAb, Map, GrpAb, Map, MapSch
            ThreeIsogenyDescentCubic(φ, α) : MapSch, Any -> Crv, MapSch
            ThreeDescentByIsogeny(E) : CrvEll -> [ Crv ], [ Map ]
            Example CrvEllQNF_ThreeDescentByIsogeny (H133E38)
            Jacobian(C) : RngMPolElt -> CrvEll
            ThreeSelmerElement(E, C) : CrvEll, RngMPolElt -> Tup
            AddCubics(cubic1, cubic2 : parameters) : RngMPolElt, RngMPolElt -> RngMPolElt
            ThreeTorsionType(E) : CrvEll -> MonStgElt
            ThreeTorsionPoints(E : parameters) : CrvEll -> Tup
            ThreeTorsionMatrices(E, C) : CrvEll, RngMPolElt -> Tup

      Six and Twelve Descent
            SixDescent(C2, C3) : CrvHyp, Crv -> Crv, MapSch
            TwelveDescent(C3, C4) : Crv, Crv -> SeqEnum, MapSch

      Nine-Descent
            NineDescent(C : parameters) : Crv -> SeqEnum, List
            NineSelmerSet(C) : Crv -> RngIntElt

      Higher 2-power Isogeny Descents
            TwoPowerIsogenyDescentRankBound(E, T : parameters) : CrvEll[FldRat], PtEll[FldRat] ) -> RngIntElt, SeqEnum, SeqEnum

      p-Isogeny Descent
            pIsogenyDescent(E,P) : CrvEll, PtEll -> RngIntElt, RngIntElt, SeqEnum, CrvEll
            pIsogenyDescent(C,phi) : Crv, MapSch -> SeqEnum, List
            FakeIsogenySelmerSet(C,phi) : Crv, MapSch -> RngIntElt
            Example CrvEllQNF_pIsogenyDescent (H133E39)
            Example CrvEllQNF_pIsogenyDescent2 (H133E40)
            Example CrvEllQNF_pIsogenyDescent3 (H133E41)

 
Bibliography

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