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, RngIntElt -> [ Tup ]
ReductionType(E, p) : CrvEll, RngIntElt -> MonStgElt
FrobeniusTraceDirect(E, p) : CrvEll, RngIntElt -> RngIntElt
TracesOfFrobenius(E, B) : CrvEll, RngIntElt -> SeqEnum
Example CrvEllQNF_frobenius-traces (H122E1)
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 (H122E2)
Complex Multiplication
HasComplexMultiplication(E) : CrvEll -> BoolElt, RngIntElt
Isogenous Curves
IsogenousCurves(E) : CrvEll[FldRat] -> SeqEnum, RngIntElt
FaltingsHeight(E) : CrvEll[FldRat] -> FldReElt
Example CrvEllQNF_isog-curves (H122E3)
Mordell--Weil Group
MordellWeilShaInformation(E: parameters) : CrvEll -> [RngIntElt], [PtEll], [Tup]
Example CrvEllQNF_mwsha-example (H122E4)
TorsionSubgroup(H) : SetPtEll -> GrpAb, Map
Rank(H: parameters) : SetPtEll -> RngIntElt
RankBounds(H: parameters) : SetPtEll -> RngIntElt, RngIntElt
MordellWeilGroup(H: parameters) : SetPtEll -> GrpAb, Map
Generators(H) : SetPtEll -> [ PtEll ]
NumberOfGenerators(H) : SetPtEll -> RngIntElt
Saturation(points, n) : [ PtEll ], RngIntElt -> [ PtEll ]
Example CrvEllQNF_MordellWeil (H122E5)
Example CrvEllQNF_Rank (H122E6)
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 (H122E7)
SilvermanBound(H) : SetPtEll -> FldPrElt
SiksekBound(H: parameters) : SetPtEll -> FldPrElt
Example CrvEllQNF_Bounds (H122E8)
IsLinearlyIndependent(P, Q) : PtEll, PtEll -> BoolElt, ModTupElt
IsLinearlyIndependent(S) : [ PtEll ] -> BoolElt, ModTupElt
ReducedBasis(S) : [ PtEll ] -> [ PtEll ]
Example CrvEllQNF_LinearIndependence (H122E9)
pAdicHeight(P, p) : PtEll, RngIntElt -> FldPadElt
pAdicRegulator(S, p) : [PtEll], RngIntElt -> FldPadElt
EisensteinTwo(E, p) : CrvEll, RngIntElt -> FldPadElt
Example CrvEllQNF_padic-height (H122E10)
Two-Descent and Two-Coverings
TwoDescent(E: parameters) : CrvEll -> [CrvHyp] , [Map]
AssociatedEllipticCurve(f) : RngUPolElt -> CrvEll, Map
Example CrvEllQNF_twodescent (H122E11)
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
Verbose Information
CasselsTatePairing(C, D) : CrvHyp, CrvHyp -> RngIntElt
CasselsTatePairing(C, D) : Crv, CrvHyp -> RngIntElt
Example CrvEllQNF_cassels-tate-example (H122E12)
Four-Descent
FourDescent(f : parameters) : RngUPolElt -> [Crv]
Example CrvEllQNF_simplefourdesc (H122E13)
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 (H122E14)
Eight-Descent
EightDescent(C : parameters) : CrvEll -> [ Crv ], [ MapSch ]
Three-Descent
ThreeDescent(E : parameters) : CrvEll -> [ Crv ], List
Example CrvEllQNF_selmer-famous-example (H122E15)
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 (H122E16)
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
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_pIsogenyDesent (H122E17)
Example CrvEllQNF_pIsogenyDescent2 (H122E18)
Example CrvEllQNF_pIsogenyDescent3 (H122E19)
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) : CrvEll[FldRat], RngIntElt -> PtEll
HeegnerPoints(E, D : parameters) : CrvEll[FldRat], RngIntElt -> Tup, PtEll
Example CrvEllQNF_Heegner (H122E20)
Example CrvEllQNF_Heegner2 (H122E21)
Example CrvEllQNF_Heegner3 (H122E22)
Example CrvEllQNF_Heegner4 (H122E23)
Example CrvEllQNF_Heegner5 (H122E24)
Analytic Information
Periods(E: parameters) : CrvEll -> [ FldComElt ]
EllipticCurveFromPeriods(om: parameters) : [ FldComElt ] -> CrvEll
RealPeriod(E: parameters) : CrvEll -> FldReElt
EllipticExponential(E, z) : CrvEll, FldComElt -> [ FldComElt ]
EllipticExponential(E, S) : CrvEll, FldRatElt -> [ FldComElt ]
EllipticLogarithm(P: parameters): PtEll[FldRat] -> FldComElt
EllipticLogarithm(E, S): CrvEll, [ FldComElt ] -> FldComElt
pAdicEllipticLogarithm(P, p: parameters): PtEll, RngIntElt -> FldLocElt
Example CrvEllQNF_ell-exp (H122E25)
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 (H122E26)
Example CrvEllQNF_conjectural-regulator (H122E27)
ModularDegree(E) : CrvEll -> RngIntElt
Example CrvEllQNF_mod-deg (H122E28)
Integral and S-integral Points
IntegralPoints(E) : CrvEll -> [ PtEll ], [ Tup ]
SIntegralPoints(E, S) : CrvEll, SeqEnum -> [ PtEll ], [ Tup ]
Example CrvEllQNF_IntegralPoints (H122E29)
Example CrvEllQNF_SIntegralPoints (H122E30)
IntegralQuarticPoints(Q) : [ RngIntElt ] -> [ SeqEnum ]
IntegralQuarticPoints(Q, P) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SIntegralQuarticPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
Example CrvEllQNF_IntegralPointsSequence (H122E31)
SIntegralLjunggrenPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SIntegralDesbovesPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
Example CrvEllQNF_Desboves (H122E32)
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) : CrvEll -> MonStgElt
Example CrvEllQNF_ecdb1 (H122E33)
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 (H122E34)
Local Invariants
Conductor(E) : CrvEll -> RngOrdIdl
BadPlaces(E) : CrvEll -> SeqEnum
BadPlaces(E, L) : CrvEll -> 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
Mordell--Weil Groups
TorsionBound(E, n) : CrvEll, RngIntElt -> RngIntElt
pPowerTorsion(E, p) : CrvEll, RngIntElt -> GrpAb, Map
TorsionSubgroup(E) : CrvEll -> GrpAb, Map
MordellWeilShaInformation(E: parameters) : CrvEll -> [RngIntElt], [PtEll], [Tup]
RankBound(E) : CrvEll -> RngIntElt
Heights
NaiveHeight(P) : PtEll -> FldPrElt
Height(P : parameters) : PtEll -> FldPrElt
HeightPairingMatrix(P : parameters) : [PtEll] -> AlgMatElt
LocalHeight(P, Pl : parameters) : PtEll, PlcNumElt -> FldPrElt
Two Descent
TwoDescent(E: parameters) : CrvEll -> [CrvHyp] , [Map] , Map
TwoCover(e) : FldNumElt -> CrvHyp, Map
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 (H122E35)
Example CrvEllQNF_selmer2 (H122E36)
Example CrvEllQNF_selmer3 (H122E37)
Example CrvEllQNF_selmer4 (H122E38)
Elliptic Curve Chabauty
Chabauty(MWmap, Ecov) : Map, MapSch -> SetEnum, RngIntElt
Chabauty(MWmap, Ecov, p) : Map, MapSch, RngIntElt -> RngIntElt, SetEnum, RngIntElt, Tup
Example CrvEllQNF_ECchabauty (H122E39)
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 (H122E40)
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
Local Invariants
Conductor(E) : CrvEll -> FldPadElt
LocalInformation(E) : CrvEll, RngOrdIdl -> Tup, CrvEll
RootNumber(E) : CrvEll -> RngIntElt
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012