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MODELS OF GENUS ONE CURVES

 
Acknowledgements
 
Introduction
 
Related Functionality
 
Creation of Genus One Models
 
Predicates on Genus One Models
 
Access Functions
 
Minimisation and Reduction
 
Genus One Models as Coverings
 
Families of Elliptic Curves with Prescribed n-Torsion
 
Transformations between Genus One Models
 
Invariants for Genus One Models
 
Covariants and Contravariants for Genus One Models
 
Examples
 
Bibliography







DETAILS

 
Introduction

 
Related Functionality

 
Creation of Genus One Models
      GenusOneModel(n, seq) : RngIntElt, [ 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
      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 (H124E1)

 
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(model : parameters) : ModelG1 -> ModelG1, Tup
      Minimise(f) : RngMPolElt -> RngMPolElt, Tup
      pMinimise(f, p) : RngMPolElt, RngIntElt -> RngMPolElt, AlgMatElt
      Reduce(model) : ModelG1 -> ModelG1, 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(model : 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

 
Examples
      Example CrvG1_visible-three-sha (H124E2)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012