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 (H128E1)
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 (H128E2)
Parametrized Structures
Isogeny(E,P) : CrvEll, Pt -> MapCrvEll
SubgroupScheme(E,P) : CrvEll, Pt -> CrvEllSubgroup
Example CrvMod_Parametrized subgroup schemes (H128E3)
Associated Structures
FunctionField(X) : CrvMod -> FldFun
jFunction(X) : CrvMod -> FldFunElt
BaseCurve(X) : CrvMod -> CrvMod, MapSch
Example CrvMod_Modular base curve (H128E4)
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 (H128E5)
Modular Curves and Quotients (Canonical Embeddings)
ModularCurveQuotient(N,A) : RngIntElt, [RngIntElt] -> Crv
Example CrvMod_x0n-quotient (H128E6)
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, RngIntElt -> SeqEnum
NewModularNonHyperellipticCurveGenus3(B) : [ModSym] -> BoolElt, RngMPolElt
NewModularNonHyperellipticCurveGenus3(F) : [RngSerPowElt] -> BoolElt, RngMPolElt
ModularNonHyperellipticCurveGenus3(F) : [RngSerPowElt] -> BoolElt, RngMPolElt
Example CrvMod_jim_ex_mod_crvs_hyp (H128E7)
Example CrvMod_jim_ex_mod_crvs_jeon (H128E8)
Example CrvMod_jim_ex_mod_crvs_jeon (H128E9)
Example CrvMod_jim_ex_mod_crvs_gen3 (H128E10)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012