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Acknowledgements
Introduction
Creation Functions
Creation of an Elliptic Curve
Creation Predicates
Changing the Base Ring
Alternative Models
Predicates on Curve Models
Twists of Elliptic Curves
Operations on Curves
Elementary Invariants
Associated Structures
Predicates on Elliptic Curves
Subgroup Schemes
Creation of Subgroup Schemes
Associated Structures
Predicates on Subgroup Schemes
Points of Subgroup Schemes
Operations on Point Sets
Creation of Point Sets
Associated Structures
Predicates on Point Sets
Operations on Points
Creation of Points
Creation Predicates
Access Operations
Associated Structures
Arithmetic
Division Points
Point Order
Predicates on Points
Weil Pairing
Polynomials
Curves over the Rationals
Local Invariants
Kodaira Symbols
Complex Multiplication
Isogenous Curves
Mordell--Weil Group
Heights and Height Pairing
Two-Descent and Two-Coverings
Two Descent Using Isogenies
Invariants
The Cassels-Tate Pairing
Four-Descent
Eight-Descent
Three-Descent
Heegner Points
Analytic Information
Integral and S-integral Points
Integral Points
S-integral Points
Elliptic Curve Database
Curves over Number Fields
Local Invariants
Complex Multiplication
Torsion Information
Heights
Selmer Groups
Mordell--Weil Group
Elliptic Curve Chabauty
Auxiliary functions for etale algebras
Morphisms
Creation Functions
Structure Operations
The Endomorphism Ring
The Automorphism Group
Predicates on Isogenies
The formal group law
Curves over p-adic Fields
Local Invariants
Bibliography
DETAILS
Introduction
Creation Functions
Creation of an Elliptic Curve
EllipticCurve([a, b]) : [ RngElt ] -> CrvEll
EllipticCurve(f) : RngUPolElt -> CrvEll
EllipticCurveFromjInvariant(j) : RngElt -> CrvEll
Example CrvEll_Creation (H103E1)
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 (H103E2)
Example CrvEll_CreationFromCurve2 (H103E3)
Creation Predicates
IsEllipticCurve([a, b]) : [ RngElt ] -> BoolElt, CrvEll
IsEllipticCurve(C) : CrvHyp -> BoolElt, CrvEll, MapIsoSch, MapIsoSch
Example CrvEll_CreationTest (H103E4)
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 (H103E5)
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 (H103E6)
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 (H103E7)
IsTwist(E, F) : CrvEll, CrvEll -> BoolElt
IsQuadraticTwist(E, F) : CrvEll, CrvEll -> BoolElt, RngElt
Example CrvEll_NonquadraticTwists (H103E8)
MinimalQuadraticTwist(E) : CrvEll -> CrvEll, RngIntElt
Example CrvEll_min_twist (H103E9)
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 (H103E10)
Example CrvEll_GenericCurve (H103E11)
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 (H103E12)
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 (H103E13)
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 (H103E14)
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 (H103E15)
Example CrvEll_PointArithmetic2 (H103E16)
Example CrvEll_GenericPoint (H103E17)
Point Order
Order(P) : PtEll -> RngIntElt
FactoredOrder(P) : PtEll -> RngIntElt
Example CrvEll_PlayWithPoints (H103E18)
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 (H103E19)
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 (H103E20)
Polynomials
DefiningPolynomial(E) : CrvEll -> RngMPolElt
DivisionPolynomial(E, n) : CrvEll, RngIntElt -> RngUPolElt, RngUPolElt, RngUPolElt
TwoTorsionPolynomial(E) : CrvEll -> RngMPolElt
Example CrvEll_DivisionPolynomial (H103E21)
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, RngIntElt -> [ Tup ]
ReductionType(E, p) : CrvEll, RngIntElt -> MonStgElt
FrobeniusTraceDirect(E, p) : CrvEll, RngIntElt -> RngIntElt
TracesOfFrobenius(E, B) : CrvEll, RngIntElt -> SeqEnum
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 CrvEll_Kodaira (H103E22)
Complex Multiplication
HasComplexMultiplication(E) : CrvEll -> BoolElt, RngIntElt
Isogenous Curves
IsogenousCurves(E) : CrvEll[FldRat] -> SeqEnum, RngIntElt
FaltingsHeight(E) : CrvEll[FldRat] -> FldReElt
Example CrvEll_isog-curves (H103E23)
Mordell--Weil Group
MordellWeilShaInformation(E: parameters) : CrvEll -> [RngIntElt], [PtEll], [Tup]
Rank(H: parameters) : SetPtEll -> RngIntElt
RankBounds(H: parameters) : SetPtEll -> RngIntElt, RngIntElt
AbelianGroup(H: parameters) : SetPtEll -> GrpAb, Map
Saturation(points, n) : [ PtEll ], RngIntElt -> [ PtEll ]
TorsionSubgroup(H) : SetPtEll -> GrpAb, Map
Generators(H) : SetPtEll -> [ PtEll ]
NumberOfGenerators(H) : SetPtEll -> RngIntElt
Example CrvEll_MordellWeil (H103E24)
Example CrvEll_Rank (H103E25)
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 CrvEll_FunWithHeights (H103E26)
SilvermanBound(H) : SetPtEll -> FldPrElt
SiksekBound(H: parameters) : SetPtEll -> FldPrElt
Example CrvEll_Bounds (H103E27)
IsLinearlyIndependent(P, Q) : PtEll, PtEll -> BoolElt, ModTupElt
IsLinearlyIndependent(S) : [ PtEll ] -> BoolElt, ModTupElt
ReducedBasis(S) : [ PtEll ] -> [ PtEll ]
Example CrvEll_LinearIndependence (H103E28)
Two-Descent and Two-Coverings
TwoDescent(E : parameters) : CrvEll -> [CrvHyp]
AssociatedEllipticCurve(f) : RngUPolElt -> CrvEll, Map
Example CrvEll_twodescent (H103E29)
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
The Cassels-Tate Pairing
CasselsTatePairing(C, D) : CrvHyp, CrvHyp -> RngIntElt
Four-Descent
FourDescent(f : parameters) : RngUPolElt -> [Crv]
Example CrvEll_simplefourdesc (H103E30)
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 CrvEll_fourdescent (H103E31)
Eight-Descent
EightDescent(C : parameters) : CrvEll -> [ Crv ], [ MapSch ]
Three-Descent
ThreeDescent(E : parameters) : CrvEll -> [ Crv ], List
Example CrvEll_selmer-famous-example (H103E32)
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
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
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 CrvEll_Heegner (H103E33)
Example CrvEll_Heegner2 (H103E34)
Example CrvEll_Heegner3 (H103E35)
Example CrvEll_Heegner4 (H103E36)
Example CrvEll_Heegner5 (H103E37)
Analytic Information
Periods(E: parameters) : CrvEll -> [ FldComElt ]
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 CrvEll_ell-exp (H103E38)
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 CrvEll_analytic-rank (H103E39)
Example CrvEll_conjectural-regulator (H103E40)
ModularDegree(E) : CrvEll -> RngIntElt
Example CrvEll_mod-deg (H103E41)
Integral and S-integral Points
Integral Points
IntegralPoints(E) : CrvEll -> [ PtEll ], [ Tup ]
Example CrvEll_IntegralPoints (H103E42)
IntegralQuarticPoints(Q) : [ RngIntElt ] -> [ SeqEnum ]
IntegralQuarticPoints(Q, P) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
Example CrvEll_IntegralPointsSequence (H103E43)
S-integral Points
SIntegralPoints(E, S) : CrvEll, SeqEnum -> [ PtEll ], [ Tup ]
Example CrvEll_SIntegralPoints (H103E44)
SIntegralQuarticPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SIntegralLjunggrenPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
SIntegralDesbovesPoints(Q, S) : [ RngIntElt ], [ RngIntElt ] -> [ SeqEnum ]
Example CrvEll_Desboves (H103E45)
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 CrvEll_ecdb1 (H103E46)
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 CrvEll_ecdb2 (H103E47)
Curves over Number Fields
Local Invariants
BadPlaces(E) : CrvEll -> SeqEnum
BadPlaces(E, L) : CrvEll -> SeqEnum
Conductor(E) : CrvEll -> RngOrdIdl
LocalInformation(E, P) : CrvEll, RngOrdIdl -> Tup, CrvEll
LocalInformation(E) : CrvEll -> [ Tup ]
Reduction(E, p) : CrvEll, RngOrdIdl -> CrvEll, Map
RootNumber(E, P) : CrvEll, RngOrdIdl -> RngIntElt
RootNumber(E) : CrvEll -> RngIntElt
Complex Multiplication
HasComplexMultiplication(E) : CrvEll -> BoolElt, RngIntElt
Torsion Information
TorsionBound(E, n) : CrvEll, RngIntElt -> RngIntElt
pPowerTorsion(E, p) : CrvEll, RngIntElt -> GrpAb, Map
TorsionSubgroup(E) : CrvEll -> GrpAb, Map
Heights
NaiveHeight(P) : PtEll -> FldPrElt
Height(P : parameters) : PtEll -> FldPrElt
HeightPairingMatrix(P : parameters) : [PtEll] -> AlgMatElt
LocalHeight(P, Pl : parameters) : PtEll, PlcNumElt -> FldPrElt
Selmer Groups
DescentMaps(phi) : Map -> Map, Map
SelmerGroup(phi) : Map -> GrpAb, Map, Map, SeqEnum, SetEnum
TwoSelmerGroup(E) : CrvEll -> GrpAb, Map, SetEnum, Map, SeqEnum
RankBound(E) : CrvEll -> RngIntElt
Example CrvEll_selmer (H103E48)
Example CrvEll_selmer2 (H103E49)
Example CrvEll_selmer3 (H103E50)
Mordell--Weil Group
MordellWeilSubgroup(E) : CrvEll -> BoolElt, GrpAb, Map
Elliptic Curve Chabauty
Chabauty(MWmap, Ecov, p) : Map, MapSch, RngIntElt -> RngIntElt, Tup, RngIntElt, SetEnum
Example CrvEll_ECchabauty (H103E51)
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 CrvEll_selmer-etale (H103E52)
Morphisms
Creation Functions
Example CrvEll_Isogeny (H103E53)
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 (H103E54)
IsIsomorphism(I) : Map -> BoolElt, Map
IsomorphismToIsogeny(I) : Map -> Map
Example CrvEll_Isomorphism (H103E55)
TranslationMap(E, P) : CrvEll, PtEll -> Map
RationalMap(i, t) : Map, Map -> Map
TwoIsogeny(P) : PtEll -> Map
Example CrvEll_Map (H103E56)
IsogenyFromKernel(G) : SchGrpEll -> CrvEll, Map
IsogenyFromKernelFactored(G) : SchGrpEll -> CrvEll, Map
IsogenyFromKernel(E, psi) : CrvEll, RngUPolElt -> CrvEll, Map
IsogenyFromKernelFactored(E, psi) : SchGrpEll -> CrvEll, Map
PushThroughIsogeny(I, v) : Map, RngUPolElt -> RngUPolElt
DualIsogeny(phi) : Map -> Map
Example CrvEll_DualIsogeny (H103E57)
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
The Endomorphism Ring
MultiplicationByMMap(E, m) : CrvEll, RngIntElt -> Map
IdentityIsogeny(E) : CrvEll -> Map
IdentityMap(E) : CrvEll -> Map
FrobeniusMap(E, i) : CrvEll, RngIntElt -> Map
FrobeniusMap(E) : CrvEll -> Map
Example CrvEll_Frobenius (H103E58)
The Automorphism Group
IdentityMap(E) : CrvEll -> Map
NegationMap(E) : CrvEll -> Map
f * g : Map, Map -> Map
Predicates on Isogenies
IsZero(I) : Map -> BoolElt
I eq J : Map, Map -> BoolElt
The formal group law
FormalGroupLaw(E, prec) : CrvEll, RngIntElt -> RngMPolElt
FormalGroupHomomorphism(phi, prec) : MapSch, RngIntElt -> RngSerPowElt
FormalLog(E) : CrvEll -> RngSerPowElt, PtEll
Curves over p-adic Fields
Local Invariants
Conductor(E) : CrvEll -> FldPadElt
LocalInformation(E) : CrvEll, RngOrdIdl -> Tup, CrvEll
Bibliography
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