Introduction

 
Creation Functions

      Creation of a Modular Curve
            ModularCurve(X,t,N) : Sch, MonStgElt, RngIntElt -> CrvMod
            ModularCurve(D, N) : DB, RngIntElt -> CrvMod

      Creation of Points
            ModuliPoints(X,E) : CrvMod, CrvEll -> SeqEnum
            Example CrvMod_Moduli points (H140E1)

 
Invariants
      Level(X) : CrvMod -> RngIntElt
      Genus(X) : CrvMod -> RngIntElt
      ModelType(X) : CrvMod -> MonStgElt
      Indices(X) : CrvMod -> SeqEnum

 
Modular Polynomial Databases
      AtkinModularPolynomial(N) : RngIntElt -> RngMPolElt
      CanonicalModularPolynomial(N) : RngIntElt -> RngMPolElt
      ClassicalModularPolynomial(N) : RngIntElt -> RngMPolElt
      ModularCurveDatabase(t) : MonStgElt -> DB
      N in D: RngIntElt, DB -> BoolElt
      ExistsModularCurveDatabase(t) : MonStgElt -> BoolElt
      Example CrvMod_Modular polynomials (H140E2)

 
Parametrized Structures
      Isogeny(E,P) : CrvEll, Pt -> MapCrvEll
      SubgroupScheme(E,P) : CrvEll, Pt -> CrvEllSubgroup
      Example CrvMod_Parametrized subgroup schemes (H140E3)

 
Associated Structures
      FunctionField(X) : CrvMod -> FldFun
      jFunction(X) : CrvMod -> FldFunElt
      BaseCurve(X) : CrvMod -> CrvMod, MapSch
      Example CrvMod_Modular base curve (H140E4)

 
Automorphisms
      CanonicalInvolution(X) : CrvMod -> MapSch

 
Class Polynomials
      HilbertClassPolynomial(D) : RngIntElt -> RngUPolElt
      WeberClassPolynomial(D) : RngIntElt -> RngUPolElt, FldFunRatUElt
      WeberToHilbertClassPolynomial(f,D) : RngUPolElt, RngIntElt -> RngUPolElt
      Example CrvMod_Standard class polynomials (H140E5)

 
Modular Curves and Quotients (Canonical Embeddings)
      ModularCurveQuotient(N,A) : RngIntElt, [RngIntElt] -> Crv
      Example CrvMod_x0n-quotient (H140E6)

 
Modular Curves of Given Level and Genus
      SetVerbose("ModularCurve", v) : MonStgElt, RngIntElt ->
      NewModularHyperellipticCurves(N, g) : RngIntElt, RngIntElt -> SeqEnum
      NewModularHyperellipticCurve(B) : [ModSym] -> BoolElt, RngUPol
      NewModularHyperellipticCurve(F) : [RngSerPowElt] -> BoolElt, RngUPol
      ModularHyperellipticCurve(B) : [ModSym] -> BoolElt, RngUPol
      ModularHyperellipticCurve(F) : [RngSerPowElt] -> BoolElt, RngUPol
      NewModularNonHyperellipticCurvesGenus3(N) : RngIntElt -> SeqEnum
      NewModularNonHyperellipticCurveGenus3(B) : [ModSym] -> BoolElt, RngMPolElt
      NewModularNonHyperellipticCurveGenus3(F) : [RngSerPowElt] -> BoolElt, RngMPolElt
      ModularNonHyperellipticCurveGenus3(F) : [RngSerPowElt] -> BoolElt, RngMPolElt
      Example CrvMod_jim_ex_mod_crvs_hyp (H140E7)
      Example CrvMod_jim_ex_mod_crvs_jeon (H140E8)
      Example CrvMod_jim_ex_mod_crvs_jeon (H140E9)
      Example CrvMod_jim_ex_mod_crvs_gen3 (H140E10)

 
Bibliography

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