Introduction

 
Creation of Genus One Models
      GenusOneModel(seq) : [ RngElt ] -> ModelG1
      GenusOneModel(C) : Crv -> ModelG1
      GenusOneModel(f) : RngMPolElt -> ModelG1
      GenusOneModel(n, E) : RngIntElt, CrvEll -> ModelG1, Crv, MapSch, MapSch
      GenusOneModel(mat) : Mtrx -> ModelG1
      GenusOneModel(mats) : [ AlgMatElt ] -> ModelG1
      IsGenusOneModel(f) : RngUPolElt -> BoolElt, ModelG1
      GenericModel(n) : RngIntElt -> ModelG1
      RandomGenusOneModel(n) : RngIntElt -> ModelG1
      ChangeRing(model, R) : ModelG1, Rng -> ModelG1
      CompleteTheSquare(model) : ModelG1 -> 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 (H135E1)

 
Attributes of Genus One Models
      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
      ModelToSequence(model) : ModelG1 -> [ RngElt ]
      ModelToString(model) : ModelG1 -> MonStgElt

 
Transformations between Genus One Models
      IsTransformation(n, g) : RngIntElt, Tup -> BoolElt, TransG1
      Tuple(g) : TransG1 -> Tup
      ChangeRing(g, R) : TransG1, Rng -> TransG1
      IdentityTransformation(n, R) : RngIntElt, Rng -> TransG1
      RandomTransformation(n : parameters) : RngIntElt -> TransG1
      g * model : TransG1, ModelG1 -> ModelG1
      g1 * g2 : TransG1, TransG1 -> TransG1
      Inverse(g) : TransG1 -> TransG1
      ScalingFactor(g) : TransG1 -> RngElt
      Determinant(g) : TransG1 -> RngElt

 
Equivalence of Genus One Models
      IsEquivalent(model1, model2) : ModelG1, ModelG1 -> BoolElt, TransG1

 
Minimisation and Reduction
      Minimise(model : parameters) : ModelG1 -> ModelG1, TransG1
      Minimise(f) : RngMPolElt -> RngMPolElt, TransG1
      pMinimise(f, p) : RngMPolElt, RngIntElt -> RngMPolElt, AlgMatElt
      Reduce(model) : ModelG1 -> ModelG1, TransG1
      ReduceQuadrics(seq) : [RngMPolElt] -> [RngMPolElt], AlgMatElt, AlgMatElt

 
Local Solubility
      IsLocallySoluble(model, p) : ModelG1, RngIntElt -> BoolElt, Any
      IsLocallySoluble(model) : ModelG1 -> BoolElt

 
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(model : parameters) : ModelG1 -> Crv, Crv, MapSch

 
Families of Elliptic Curves with Prescribed n-Torsion
      RubinSilverbergPolynomials(n, J : parameters) : RngIntElt, RngElt -> RngElt, 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, RngIntElt -> ModelG1, ModelG1
      HessePolynomials(n, r, invariants : parameters) : RngIntElt, RngIntElt, [RngElt] -> RngElt, RngElt, RngElt

 
Examples
      Example CrvG1_visible-three-sha (H135E2)

 
Bibliography

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