- 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 (H130E1)
- Kodaira Symbols
- Complex Multiplication
- Isogenous Curves
- 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 (H130E4)
- SilvermanBound(H) : SetPtEll -> FldPrElt
- SiksekBound(H: parameters) : SetPtEll -> FldPrElt
- Example CrvEllQNF_Bounds (H130E5)
- IsLinearlyIndependent(P, Q) : PtEll, PtEll -> BoolElt, ModTupElt
- IsLinearlyIndependent(S) : [ PtEll ] -> BoolElt, ModTupElt
- ReducedBasis(S) : [ PtEll ] -> [ PtEll ]
- Example CrvEllQNF_LinearIndependence (H130E6)
- 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 (H130E7)
- 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 (H130E8)
- Example CrvEllQNF_Heegner2 (H130E9)
- Example CrvEllQNF_Heegner3 (H130E10)
- Example CrvEllQNF_Heegner4 (H130E11)
- Example CrvEllQNF_Heegner5 (H130E12)
- 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 (H130E13)
- Example CrvEllQNF_ellexp-nf (H130E14)
- 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 (H130E15)
- Example CrvEllQNF_conjectural-regulator (H130E16)
- ModularDegree(E) : CrvEll -> RngIntElt
- Example CrvEllQNF_mod-deg (H130E17)
- Integral and S-integral Points
- IntegralPoints(E) : CrvEll[FldRat] -> [ PtEll ]
- SIntegralPoints(E, S) : CrvEll, SeqEnum -> [ PtEll ]
- Example CrvEllQNF_IntegralPoints (H130E18)
- Example CrvEllQNF_SIntegralPoints (H130E19)
- IntegralQuarticPoints(Q) : [ RngIntElt ] -> [ SeqEnum ]
- IntegralQuarticPoints(Q, P) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
- SIntegralQuarticPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
- Example CrvEllQNF_IntegralPointsSequence (H130E20)
- SIntegralLjunggrenPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
- SIntegralDesbovesPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
- Example CrvEllQNF_Desboves (H130E21)
- 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 (H130E22)
- 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 (H130E23)
- Curves over Number Fields
- Local Invariants
- Complex Multiplication
- Heights
- Integral Points
- Elliptic Curve Chabauty
- Chabauty(MWmap, Ecov) : Map, MapSch -> SetEnum, RngIntElt
- Chabauty(MWmap, Ecov, p) : Map, MapSch, RngIntElt -> RngIntElt, SetEnum, RngIntElt, Tup
- Example CrvEllQNF_ECchabauty (H130E24)
- Auxiliary Functions for Etale Algebras
- Analytic Information
- Elliptic Curves of Given Conductor
- Curves over p-adic Fields
- Mordell-Weil Groups and Descent Methods
- Torsion
- Mordell-Weil Group and Rank
- Two-Descent
- Selmer Groups
- The Cassels-Tate Pairing
- Four-Descent
- Eight-Descent
- Three-Descent and Five-Descent
- ThreeDescent(E : parameters) : CrvEll -> [ Crv ], List
- Example CrvEllQNF_selmer-famous-example (H130E37)
- 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 (H130E38)
- 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
- Nine-Descent
- Higher 2-power Isogeny Descents
- 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 (H130E39)
- Example CrvEllQNF_pIsogenyDescent2 (H130E40)
- Example CrvEllQNF_pIsogenyDescent3 (H130E41)
- Bibliography
V2.28, 13 July 2023