Creation of an Elliptic Curve
EllipticCurve([a, b]) : [ RngElt ] -> CrvEll
EllipticCurve(f) : RngUPolElt -> CrvEll
EllipticCurveFromjInvariant(j) : RngElt -> CrvEll
Example CrvEll_Creation (H120E1)
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 (H120E2)
Example CrvEll_CreationFromCurve2 (H120E3)
Creation Predicates
IsEllipticCurve([a, b]) : [ RngElt ] -> BoolElt, CrvEll
IsEllipticCurve(C) : CrvHyp -> BoolElt, CrvEll, MapIsoSch, MapIsoSch
Example CrvEll_CreationTest (H120E4)
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 (H120E5)
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 (H120E6)
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 (H120E7)
IsTwist(E, F) : CrvEll, CrvEll -> BoolElt
IsQuadraticTwist(E, F) : CrvEll, CrvEll -> BoolElt, RngElt
Example CrvEll_NonquadraticTwists (H120E8)
MinimalQuadraticTwist(E) : CrvEll -> CrvEll, RngIntElt
Example CrvEll_min_twist (H120E9)
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 (H120E10)
Example CrvEll_GenericCurve (H120E11)
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 (H120E12)
Polynomials
DefiningPolynomial(E) : CrvEll -> RngMPolElt
DivisionPolynomial(E, n) : CrvEll, RngIntElt -> RngUPolElt, RngUPolElt, RngUPolElt
TwoTorsionPolynomial(E) : CrvEll -> RngMPolElt
Example CrvEll_DivisionPolynomial (H120E13)
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 (H120E14)
The Formal Group
FormalGroupLaw(E, prec) : CrvEll, RngIntElt -> RngMPolElt
FormalGroupHomomorphism(phi, prec) : MapSch, RngIntElt -> RngSerPowElt
FormalLog(E) : CrvEll -> RngSerPowElt, PtEll
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 (H120E15)
Creation Functions
Example CrvEll_Isogeny (H120E16)
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 (H120E17)
IsIsomorphism(I) : Map -> BoolElt, Map
IsomorphismToIsogeny(I) : Map -> Map
Example CrvEll_Isomorphism (H120E18)
TranslationMap(E, P) : CrvEll, PtEll -> Map
RationalMap(i, t) : Map, Map -> Map
TwoIsogeny(P) : PtEll -> Map
Example CrvEll_Map (H120E19)
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 (H120E20)
Predicates on Isogenies
IsZero(I) : Map -> BoolElt
I eq J : Map, Map -> BoolElt
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
Endomorphisms
MultiplicationByMMap(E, m) : CrvEll, RngIntElt -> Map
IdentityIsogeny(E) : CrvEll -> Map
IdentityMap(E) : CrvEll -> Map
NegationMap(E) : CrvEll -> Map
FrobeniusMap(E, i) : CrvEll, RngIntElt -> Map
FrobeniusMap(E) : CrvEll -> Map
Example CrvEll_Frobenius (H120E21)
Automorphisms
AutomorphismGroup(E) : CrvEll -> Grp, Map
Automorphisms(E) : CrvEll -> SeqEnum
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 (H120E22)
Example CrvEll_PointArithmetic2 (H120E23)
Example CrvEll_GenericPoint (H120E24)
Point Order
Order(P) : PtEll -> RngIntElt
FactoredOrder(P) : PtEll -> RngIntElt
Example CrvEll_PlayWithPoints (H120E25)
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 (H120E26)
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 (H120E27)
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013