Introduction
Related functionality
Creation of Genus One Models GenusOneModel(n,seq) : RngIntElt, [RngElt] -> ModelG1 GenusOneModel(C) : Crv -> ModelG1 GenusOneModel(n,E) : RngIntElt, CrvEll -> ModelG1, Crv, MapSch, MapSch GenusOneModel(mat) : Mtrx -> ModelG1 GenusOneModel(mats) : SeqEnum -> ModelG1 CompleteTheSquare(model) : ModelG1 -> ModelG1 RandomGenusOneModel(n) : RngIntElt -> ModelG1 GenericModel(n) : RngIntElt -> ModelG1 ChangeRing(model,B) : ModelG1, Rng -> ModelG1 CubicFromPoint(E, P) : CrvEll, PtEll -> RngMPolElt, MapSch, Pt HesseModel(n, seq) : RngIntElt, [ RngElt ] -> ModelG1 DiagonalModel(n, seq) : RngIntElt, [ RngElt ] -> ModelG1 Example CrvG1_generic-model (H106E1)
Predicates on Genus One Models IsGenusOneModel(f) : RngMPolElt -> BoolElt, ModelG1 IsEquivalent(model1,model2) : ModelG1, ModelG1 -> BoolElt, Tup
Access Functions Degree(model) : ModelG1 -> RngIntElt DefiningEquations(model) : ModelG1 -> [ RngMPolElt ] Equations(model) : ModelG1 -> [ RngMPolElt ] Matrix(model) : ModelG1 -> Mtrx Curve(model) : ModelG1 -> Crv Matrices(model) : ModelG1 -> [ AlgMatElt ] BaseRing(model) : ModelG1 -> Rng PolynomialRing(model) : ModelG1 -> RngMPol Eltseq(model) : ModelG1 -> [ RngElt ]
Minimisation and Reduction Minimise(C) : RngMPolElt -> RngMPolElt, Tup pMinimise(C, p) : RngMPolElt, RngIntElt -> RngMPolElt, AlgMatElt Reduce(C) : RngMPolElt -> RngMPolElt, Tup ReduceQuadrics(seq) : [RngMPolElt] -> [RngMPolElt], AlgMatElt, AlgMatElt
Genus One Models as Coverings Jacobian(model) : ModelG1 -> CrvEll nCovering(model : parameters) : ModelG1 -> Crv, CrvEll, MapSch AddCubics(cubic1, cubic2 : parameters) : RngMPolElt, RngMPolElt -> RngMPolElt DoubleGenusOneModel(model) : ModelG1 -> ModelG1 FourToTwoCovering(M : parameters) : ModelG1 -> Crv, Crv, MapSch
Families of Elliptic Curves with Prescribed n-Torsion RubinSilverbergPolynomials(n, J : parameters) : RngIntElt, RngElt -> RngElt, RngElt
Transformations between Genus One Models IsTransformation(n,g) : RngIntElt, Tup -> BoolElt, RngElt RandomTransformation(n : parameters) : RngIntElt -> Tup g * model : Tup, ModelG1 -> ModelG1 g1 * g2 : Tup, Tup -> Tup ScalingFactor(g) : Tup -> RngElt
Invariants for Genus One Models aInvariants(model) : ModelG1 -> [ RngElt ] bInvariants(model) : ModelG1 -> [ RngElt ] cInvariants(model) : ModelG1 -> [ RngElt ] Invariants(model) : ModelG1 -> RngElt, RngElt, RngElt Discriminant(model) : ModelG1 -> RngElt SL4Invariants(model) : ModelG1 -> [ RngElt ]
Covariants and Contravariants for Genus One Models Hessian(model) : ModelG1 -> ModelG1 CoveringCovariants(model) : ModelG1 -> [ RngMPolElt ] Contravariants(model) : ModelG1 -> ModelG1, ModelG1 HesseCovariants(model,r) : ModelG1 -> ModelG1, ModelG1 HessePolynomials(n, r, invariants : parameters) : RngIntElt, RngIntElt, [RngElt] -> RngElt, RngElt, RngElt
Extended Example
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