[Next][Prev] [_____] [Left] [Up] [Index] [Root]

BINARY QUADRATIC FORMS

 
Acknowledgements
 
Introduction
 
Creation Functions
      Creation of Structures
      Creation of Forms
 
Basic Invariants
 
Operations on Forms
      Arithmetic
      Attribute Access
      Boolean Operations
      Related Structures
 
Class Group
 
Class Group Coercions
 
Discrete Logarithms
 
Elliptic and Modular Invariants
 
Class Invariants
 
Matrix Action on Forms
 
Bibliography







DETAILS

 
Introduction

 
Creation Functions

      Creation of Structures
            BinaryQuadraticForms(D) : RngIntElt -> QuadBin

      Creation of Forms
            Identity(Q) : QuadBin -> QuadBinElt
            Q ! [a, b, c] : QuadBin, RngIntElt, RngIntElt, RngIntElt -> QuadBinElt
            PrimeForm(Q, p) : QuadBin, RngIntElt -> QuadBinElt

 
Basic Invariants
      Discriminant(f) : QuadBinElt -> RngIntElt
      Discriminant(Q) : QuadBin -> RngIntElt
      IsDiscriminant(D) : RngIntElt -> BoolElt
      FundamentalDiscriminant(D) : RngIntElt -> RngIntElt
      IsFundamental(D) : RngIntElt -> BoolElt
      Conductor(Q) : QuadBin -> RngIntElt

 
Operations on Forms

      Arithmetic
            Conjugate(f) : QuadBinElt -> QuadBinElt
            f * g : QuadBinElt, QuadBinElt -> QuadBinElt
            f ^ n : QuadBinElt, RngIntElt -> QuadBinElt
            Reduction(f) : QuadBinElt -> QuadBinElt, Mtrx
            ReductionStep(f) : QuadBinElt -> QuadBinElt
            ReductionOrbit(f) : QuadBinElt -> SeqEnum[QuadBinElt]
            Order(f) : QuadBinElt -> RngIntElt

      Attribute Access
            f[i] : QuadBinElt, RngIntElt -> RngIntElt
            Eltseq(f) : QuadBinElt -> SeqEnum[RngIntElt]

      Boolean Operations
            f in Q : QuadBinElt, QuadBin -> BoolElt
            f eq g : QuadBinElt, QuadBinElt -> BoolElt
            IsIdentity(f) : QuadBinElt -> BoolElt
            IsReduced(f) : QuadBinElt -> BoolElt
            IsEquivalent(f, g) : QuadBinElt, QuadBinElt -> BoolElt, AlgMatElt

      Related Structures
            QuadraticOrder(Q) : QuadBin -> RngQuad
            Ideal(f) : QuadBinElt -> RngQuadIdl

 
Class Group
      ReducedForms(Q) : QuadBin -> [ QuadBinElt ]
      ReducedOrbits(Q) : QuadBin -> [ {@ QuadBinElt @} ]
      ClassNumber(Q: parameters) : QuadBin -> RngIntElt
      ClassGroup(Q: parameters) : QuadBin -> GrpAb, Map
      ClassGroupStructure(Q: parameters) : QuadBin -> [ RngIntElt ]
      AmbiguousForms(Q) : QuadBin -> SeqEnum
      TwoTorsionSubgroup(Q) : QuadBin -> GrpAb, Map
      Example QuadBin_Forms (H33E1)

 
Class Group Coercions
      FundamentalQuotient(Q) : QuadBin -> Map
      QuotientMap(Q1, Q2) : QuadBin, QuadBin -> Map
      Q ! f : QuadBin, QuadBinElt -> QuadBinElt

 
Discrete Logarithms
      Log(b, x): QuadBinElt, QuadBinElt -> RngIntElt
      Log(b, x, t): QuadBinElt, QuadBinElt, RngIntElt -> RngIntElt

 
Elliptic and Modular Invariants
      Lattice(f) : QuadBinElt -> Lat
      GramMatrix(f) : QuadBinElt -> AlgMatElt
      ThetaSeries(f, n) : QuadBinElt, RngIntElt -> RngSerElt
      RepresentationNumber(f, n) : QuadBinElt, RngIntElt -> RngIntElt
      jInvariant(f) : QuadBinElt -> RngSerElt
      Eisenstein(k, f) : RngIntElt, QuadBinElt -> RngSerElt
      WeierstrassSeries(z, f) : RngSerElt, QuadBinElt -> RngSerElt
      Example QuadBin_ell_mod (H33E2)

 
Class Invariants
      HilbertClassPolynomial(D) : RngIntElt -> RngUPolElt
      WeberClassPolynomial(D) : RngIntElt -> RngUPolElt

 
Matrix Action on Forms
      f * M : QuadBinElt, GrpMatElt -> QuadBinElt

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Mon Dec 17 14:40:36 EST 2012