Elliptic Curves over \({\mathbb{Q}}\) and Number Fields
- Introduction
- Curves over the Rationals
- Local Invariants
Conductor(E): CrvEll → RngIntEltBadPrimes(E): CrvEll → [ RngIntElt ]TamagawaNumber(E, p): CrvEll, RngIntElt → RngIntEltTamagawaNumbers(E): CrvEll → [ RngIntElt ]LocalInformation(E, p): CrvEll, RngIntElt → <RngIntElt, RngIntElt, RngIntElt, RngIntElt, SymKod, BoolElt>, CrvEllLocalInformation(E): CrvEll → [ Tup ]ReductionType(E, p): CrvEll, RngIntElt → MonStgEltTraceOfFrobeniusDirect(E, p): CrvEll, RngIntElt → RngIntEltTracesOfFrobenius(E, B): CrvEll, RngIntElt → SeqEnumExample: Frobenius Traces
- Kodaira Symbols
- Complex Multiplication
- Isogenous Curves
- Heights and Height Pairing
NaiveHeight(P): PtEll → FldPrEltWeilHeight(P): PtEll → FldPrEltHeight(P: parameters): PtEll → NFldComEltCanonicalHeight(P: parameters): PtEll → NFldComEltLocalHeight(P, p): PtEll, RngIntElt → FldComEltHeightPairing(P, Q: parameters): PtEll, PtEll → FldComEltHeightPairingMatrix(S: parameters): [PtEll] → AlgMatHeightPairingMatrix(E: parameters): CrvEll → AlgMatRegulator(S): [ PtEll ] → FldComEltRegulator(E): CrvEll → FldComEltExample: Fun With HeightsSilvermanBound(H): SetPtEll → FldPrEltSilvermanBound(E): CrvEll → FldPrEltSiksekBound(H: parameters): SetPtEll → FldPrEltSiksekBound(E: parameters): CrvEll → FldPrEltExample: BoundsIsLinearlyIndependent(P, Q): PtEll, PtEll → BoolElt, ModTupEltIsLinearlyIndependent(S): [ PtEll ] → BoolElt, ModTupEltReducedBasis(S): [ PtEll ] → [ PtEll ]Example: Linear IndependencepAdicHeight(P, p): PtEll, RngIntElt → FldPadEltpAdicRegulator(S, p): [PtEll], RngIntElt → FldPadEltEisensteinTwo(E, p): CrvEll, RngIntElt → FldPadEltFrobeniusMatrix(E, p): CrvEll, RngIntElt → MtrxExample: Padic Height
- Heegner Points
HeegnerPoint(E : parameters): CrvEll → BoolElt, PtEllHeegnerPoint(C : parameters): CrvHyp → BoolElt, PtHypHeegnerPoint(f : parameters): RngUPolElt → BoolElt, PtHypHeegnerPoint(C : parameters): Crv → BoolElt, PtModularParametrization(E, z, B : parameters): CrvEll[FldRat], FldComElt, RngIntElt → FldComEltModularParametrisation(E, z, B : parameters): CrvEll[FldRat], FldComElt, RngIntElt → FldComEltModularParametrization(E, z : parameters): CrvEll[FldRat], FldComElt → FldComEltModularParametrisation(E, z : parameters): CrvEll[FldRat], FldComElt → FldComEltModularParametrization(E, Z, B : parameters): CrvEll[FldRat], [FldComElt], RngIntElt → [FldComElt]ModularParametrisation(E, Z, B : parameters): CrvEll[FldRat], [FldComElt], RngIntElt → [FldComElt]ModularParametrization(E, Z : parameters): CrvEll[FldRat], [FldComElt] → [FldComElt]ModularParametrisation(E, Z : parameters): CrvEll[FldRat], [FldComElt] → [FldComElt]ModularParametrization(E, f, B : parameters): CrvEll[FldRat], QuadBinElt, RngIntElt → FldComEltModularParametrisation(E, f, B : parameters): CrvEll[FldRat], QuadBinElt, RngIntElt → FldComEltModularParametrization(E, f : parameters): CrvEll[FldRat], QuadBinElt → FldComEltModularParametrisation(E, f : parameters): CrvEll[FldRat], QuadBinElt → FldComEltModularParametrization(E, F, B : parameters): CrvEll[FldRat], [QuadBinElt], RngIntElt → [FldComElt]ModularParametrisation(E, F, B : parameters): CrvEll[FldRat], [QuadBinElt], RngIntElt → [FldComElt]ModularParametrization(E, F : parameters): CrvEll[FldRat], [QuadBinElt] → [FldComElt]ModularParametrisation(E, F : parameters): CrvEll[FldRat], [QuadBinElt] → [FldComElt]HeegnerDiscriminants(E,lo,hi): CrvEll[FldRat], RngIntElt, RngIntElt → SeqEnumHeegnerForms(E,D : parameters): CrvEll[FldRat], RngIntElt → SeqEnumHeegnerForms(N,D : parameters): RngIntElt, RngIntElt → SeqEnumManinConstant(E): CrvEll[FldRat] → RngIntEltHeegnerTorsionElement(E, Q): CrvEll[FldRat], RngIntElt → PtEllHeegnerPoints(E, D : parameters): CrvEll[FldRat], RngIntElt → Tup, PtEllExample: HeegnerExample: Heegner2Example: Heegner3Example: Heegner4Example: Heegner5
- Analytic Information
Periods(E: parameters): CrvEll → [ FldComElt ]Periods(E, k): CrvEll, RngIntElt → [ FldComElt ]EllipticCurveFromPeriods(om: parameters): [ FldComElt ] → CrvEllRealPeriod(E: parameters): CrvEll → FldReEltEllipticExponential(E, z): CrvEll, FldComElt → [ FldComElt ]EllipticExponential(E, k, z): CrvEll, RngIntElt, FldComElt → [ FldComElt ]EllipticExponential(E, S): CrvEll, [ FldRat ] → [ FldComElt ]EllipticLogarithm(P): PtEll[FldRat] → FldComEltEllipticLogarithm(P, k): PtEll[FldNum], RngIntElt → FldComEltEllipticLogarithm(E, S): CrvEll, [ FldComElt ] → FldComEltpAdicEllipticLogarithm(P, p: parameters): PtEll, RngIntElt → FldLocEltExample: Ell ExpExample: Ellexp NfRootNumber(E): CrvEll → RngIntEltRootNumber(E, p): CrvEll, RngIntElt → RngIntEltAnalyticRank(E): CrvEll → RngIntElt, FldReEltConjecturalRegulator(E): CrvEll → FldReElt, RngIntEltConjecturalRegulator(E, v): CrvEll, FldReElt → FldReEltExample: Analytic RankExample: Conjectural RegulatorModularDegree(E): CrvEll → RngIntEltExample: Mod Deg
- Integral and \(S\)-integral Points
IntegralPoints(E): CrvEll[FldRat] → [ PtEll ]SIntegralPoints(E, S): CrvEll, SeqEnum → [ PtEll ]Example: Integral PointsExample: S Integral PointsIntegralQuarticPoints(Q): [ RngIntElt ] → [ SeqEnum ]IntegralQuarticPoints(Q, P): [ RngIntElt ], [ RngIntElt ] → [ SeqEnum ]SIntegralQuarticPoints(Q, S): [ RngIntElt ], [ RngIntElt ] → [ SeqEnum ]Example: Integral Points SequenceSIntegralLjunggrenPoints(Q, S): [ RngIntElt ], [ RngIntElt ] → [ SeqEnum ]SIntegralDesbovesPoints(Q, S): [ RngIntElt ], [ RngIntElt ] → [ SeqEnum ]Example: Desboves
- Elliptic Curve Database
EllipticCurveDatabase(: parameters) → DBCremonaDatabase(: parameters) → DBSetBufferSize(D, n): DB, RngIntEltLargestConductor(D): DB → RngIntEltConductorRange(D): DB → RngIntElt, RngIntElt# D: DB → RngIntEltNumberOfCurves(D): DB → RngIntEltNumberOfCurves(D, N): DB, RngIntElt → RngIntEltNumberOfCurves(D, N, i): DB, RngIntElt, RngIntElt → RngIntEltNumberOfIsogenyClasses(D, N): DB, RngIntElt → RngIntEltEllipticCurve(D, N, I, J): DB, RngIntElt, RngIntElt, RngIntElt → CrvEllEllipticCurve(D, N, S, J): DB, RngIntElt, MonStgElt, RngIntElt → CrvEllEllipticCurve(D, S): DB, MonStgElt → CrvEllEllipticCurve(S): MonStgElt → CrvEllRandom(D): DB → CrvEllCremonaReference(D, E): DB, CrvEll → MonStgEltCremonaReference(E): CrvEll → MonStgEltExample: ecdb1EllipticCurves(D, N, I): DB, RngIntElt, RngIntElt → [ CrvEll ]EllipticCurves(D, N, S): DB, RngIntElt, MonStgElt → [ CrvEll ]EllipticCurves(D, N): DB, RngIntElt → [ CrvEll ]EllipticCurves(D, S): DB, MonStgElt → [ CrvEll ]EllipticCurves(D): DB → [ CrvEll ]Example: ecdb2
- Curves over Number Fields
- Local Invariants
- Complex Multiplication
- Heights
- Integral Points
- Elliptic Curve Chabauty
Chabauty(MWmap, Ecov): Map, MapSch → SetEnum, RngIntEltChabauty(MWmap, Ecov, p): Map, MapSch, RngIntElt → RngIntElt, SetEnum, RngIntElt, TupExample: E Cchabauty
- 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
RankBounds(H: parameters): SetPtEll → RngIntElt, RngIntEltRankBounds(E: parameters): CrvEll → RngIntElt, RngIntEltMordellWeilRankBounds(H: parameters): SetPtEll → RngIntElt, RngIntEltMordellWeilRankBounds(E: parameters): CrvEll → RngIntElt, RngIntEltRank(H: parameters): SetPtEll → RngIntElt, BoolEltRank(E: parameters): CrvEll → RngIntElt, BoolEltRankBound(E): CrvEll → RngIntElt, BoolEltMordellWeilRank(H: parameters): SetPtEll → RngIntElt, BoolEltMordellWeilRank(E: parameters): CrvEll → RngIntElt, BoolEltMordellWeilGroup(H: parameters): SetPtEll → GrpAb, Map, BoolElt, BoolEltMordellWeilGroup(E: parameters): CrvEll → GrpAb, Map, BoolElt, BoolEltAbelianGroup(H: parameters): SetPtEll → GrpAb, Map, BoolElt, BoolEltAbelianGroup(E: parameters): CrvEll → GrpAb, Map, BoolElt, BoolEltGenerators(H): SetPtEll → [ PtEll ]Generators(E): CrvEll → [ PtEll ]NumberOfGenerators(H): SetPtEll → RngIntEltNumberOfGenerators(E): CrvEll → RngIntEltNgens(H): SetPtEll → RngIntEltNgens(E): CrvEll → RngIntEltSaturation(points, n): [ PtEll ], RngIntElt → [ PtEll ]Saturation(points): [ PtEll ] → [ PtEll ]Example: Mordell WeilExample: RankMordellWeilShaInformation(E: parameters): CrvEll → [RngIntElt], [PtEll], [Tup]DescentInformation(E: parameters): CrvEll → [RngIntElt], [PtEll], [Tup]Example: Mwsha Example
- Two-Descent
- Selmer Groups
DescentMaps(phi): Map → Map, MapCasselsMap(phi): Map → Map, MapSelmerGroup(phi): Map → GrpAb, Map, Map, SeqEnum, SetEnumTwoSelmerGroup(E): CrvEll → GrpAb, Map, SetEnum, Map, SeqEnumExample: selmerExample: selmer2Example: selmer3Example: selmer4
- The Cassels-Tate Pairing
- Four-Descent
FourDescent(C : parameters): CrvHyp → [Crv]FourDescent(f : parameters): RngUPolElt → [Crv]FourDescent(S : parameters): SeqEnum → [Crv]FourDescent(C : parameters): ModelG1 → [Crv]Example: simplefourdescAssociatedEllipticCurve(qi): Crv → CrvEll, MapAssociatedHyperellipticCurve(qi): Crv → CrvHyp, MapQuadricIntersection(F): [AlgMatElt] → CrvQuadricIntersection(P, F): Prj, [AlgMatElt] → CrvQuadricIntersection(E): CrvEll → Crv, MapIsoSchQuadricIntersection(C): CrvHyp → Crv, MapIsoSchIsQuadricIntersection(C): Crv → BoolElt, [AlgMatElt]PointsQI(C, B : parameters): Crv, RngIntElt → [Pt]TwoCoverPullback(H, pt): CrvHyp[FldRat], PtEll[FldRat] → [PtHyp]TwoCoverPullback(f, pt): RngUPolElt[FldRat], PtEll[FldRat] → [PtHyp]FourCoverPullback(C, pt): Crv[FldRat], PtEll[FldRat] → [Pt]FourCoverPullback(C, pt): Crv[FldRat], PtHyp[FldRat] → [Pt]Example: fourdescent
- Eight-Descent
- Three-Descent and Five-Descent
ThreeDescent(E : parameters): CrvEll → [ Crv ], ListExample: Selmer Famous ExampleThreeSelmerGroup(E : parameters): CrvEll → GrpAb, MapThreeDescentCubic(E, α : parameters): CrvEll, Tup → Crv, MapSchThreeIsogenyDescent(E : parameters): CrvEll → [ Crv ], List, [ Crv ], List, MapSchThreeIsogenySelmerGroups(E : parameters): CrvEll → GrpAb, Map, GrpAb, Map, MapSchThreeIsogenyDescentCubic(φ, α): MapSch, Any → Crv, MapSchThreeDescentByIsogeny(E): CrvEll → [ Crv ], [ Map ]Example: Three Descent By IsogenyJacobian(C): RngMPolElt → CrvEllThreeSelmerElement(E, C): CrvEll, RngMPolElt → TupThreeSelmerElement(E, C): CrvEll, Crv → TupThreeSelmerElement(C): RngMPolElt → TupThreeSelmerElement(C): Crv → TupAddCubics(cubic1, cubic2 : parameters): RngMPolElt, RngMPolElt → RngMPolEltThreeTorsionType(E): CrvEll → MonStgEltThreeTorsionPoints(E : parameters): CrvEll → TupThreeTorsionMatrices(E, C): CrvEll, RngMPolElt → Tup
- Six and Twelve Descent
SixDescent(C2, C3): CrvHyp, Crv → Crv, MapSchSixDescent(model2, model3): ModelG1, ModelG1 → Crv, MapSchTwelveDescent(C3, C4): Crv, Crv → SeqEnum, MapSchTwelveDescent(model3, model4): ModelG1, ModelG1 → SeqEnum, MapSch
- Nine-Descent
- Higher 2-power Isogeny Descents
- \(p\)-Isogeny Descent
pIsogenyDescent(E,P): CrvEll, PtEll → RngIntElt, RngIntElt, SeqEnum, CrvEllpIsogenyDescent(E,p): CrvEll, RngIntElt → RngIntElt, RngIntElt, SeqEnum, CrvEllpIsogenyDescent(lambda,p): FldRatElt, RngIntElt → RngIntElt, RngIntElt, SeqEnum, CrvEll, CrvEllpIsogenyDescent(C,phi): Crv, MapSch → SeqEnum, ListpIsogenyDescent(C,E1,E2): Crv, CrvEll, CrvEll → SeqEnum, ListpIsogenyDescent(C,P): Crv, PtEll → SeqEnum, ListFakeIsogenySelmerSet(C,phi): Crv, MapSch → RngIntEltFakeIsogenySelmerSet(C,E1,E2): Crv, CrvEll, CrvEll → RngIntEltFakeIsogenySelmerSet(C,P): Crv, PtEll → RngIntEltExample: p Isogeny DescentExample: p Isogeny Descent2Example: p Isogeny Descent3
