Binary Quadratic Forms

• Introduction
• Creation Functions
  • Creation of Structures
    • BinaryQuadraticForms(D): RngIntElt → QuadBin
    • QuadraticForms(D): RngIntElt → QuadBin
  • Creation of Forms
    • Identity(Q): QuadBin → QuadBinElt
    • Q ! 1: QuadBin, RngIntElt → QuadBinElt
    • Q ! [a, b, c]: QuadBin, RngIntElt, RngIntElt, RngIntElt → QuadBinElt
    • elt< Q | a, b, c>: QuadBin, RngIntElt, RngIntElt, RngIntElt → QuadBinElt
    • elt< Q | a, b>: QuadBin, RngIntElt, RngIntElt → QuadBinElt
    • PrimeForm(Q, p): QuadBin, RngIntElt → QuadBinElt
• Basic Invariants
  • Discriminant(f): QuadBinElt → RngIntElt
  • Discriminant(Q): QuadBin → RngIntElt
  • IsDiscriminant(D): RngIntElt → BoolElt
  • IsFundamental(D): RngIntElt → BoolElt
  • IsFundamentalDiscriminant(D): RngIntElt → BoolElt
  • FundamentalDiscriminant(D): RngIntElt → RngIntElt
  • Conductor(Q): QuadBin → RngIntElt
• Operations on Forms
  • Arithmetic
    • Conjugate(f): QuadBinElt → QuadBinElt
    • f * g: QuadBinElt, QuadBinElt → QuadBinElt
    • Composition(f, g): QuadBinElt, QuadBinElt → QuadBinElt
    • f ^ n: QuadBinElt, RngIntElt → QuadBinElt
    • Power(f, n): QuadBinElt, RngIntElt → QuadBinElt
  • Matrix Action
    • f * M: QuadBinElt, GrpMatElt → QuadBinElt
  • Reduction
    • Reduction(f): QuadBinElt → QuadBinElt, Mtrx
    • ReducedForm(f): QuadBinElt → QuadBinElt, Mtrx
    • ReductionStep(f): QuadBinElt → QuadBinElt
    • ReductionOrbit(f): QuadBinElt → {@ QuadBinElt @}
    • Order(f): QuadBinElt → RngIntElt
  • Attribute Access
    • f[i]: QuadBinElt, RngIntElt → RngIntElt
    • Eltseq(f): QuadBinElt → SeqEnum[RngIntElt]
    • ElementToSequence(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
  • Maps of Forms
    • FundamentalQuotient(Q): QuadBin → Map
    • QuotientMap(Q1, Q2): QuadBin, QuadBin → Map
    • Q ! f: QuadBin, QuadBinElt → QuadBinElt
  • Related Structures
    • Parent(f): QuadBinElt → QuadBin
    • Category(Q): QuadBin → Cat
    • QuadraticOrder(Q): QuadBin → RngQuad
    • Ideal(f): QuadBinElt → RngQuadIdl
• Reduced Forms
  • ReducedForms(Q): QuadBin → [ QuadBinElt ]
  • ReducedForms(D): RngIntElt → [ QuadBinElt ]
  • ReducedOrbits(Q): QuadBin → [ {@ QuadBinElt @} ]
• Class Groups
  • ClassNumber(Q: parameters): QuadBin → RngIntElt
  • ClassNumber(D: parameters): RngIntElt → RngIntElt
  • ClassGroup(Q: parameters): QuadBin → GrpAb, Map
  • ClassGroupStructure(Q: parameters): QuadBin → [ RngIntElt ]
  • AmbiguousForms(Q): QuadBin → SeqEnum
  • TwoTorsionSubgroup(Q): QuadBin → GrpAb, Map
  • Example: Forms
• 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: Ell Mod
• Class Invariants