Rational Curve and Conic Creation
Conic(X,f) : Prj, RngMPolElt -> CrvCon
RationalCurve(X,f) : Prj, RngMPolElt -> CrvRat
IsConic(C) : Sch -> BoolElt, CrvCon
IsRationalCurve(C) : Sch -> BoolElt, CrvRat
Example CrvCon_ConicCreation (H86E1)
Access Functions
DefiningPolynomial(C) : Crv -> RngMPolElt
DefiningIdeal(C) : Crv -> RngMPol
Rational Curve and Conic Examples
Example CrvCon_RationalCurveExample (H86E2)
Example CrvCon_ConicCurve (H86E3)
Elementary invariants
Discriminant(C) : CrvCon -> FldElt
Alternative defining polynomials
LegendrePolynomial(C) : CrvCon -> RngMPolElt, ModMatRngElt
ReducedLegendrePolynomial(C) : CrvCon -> RngMPolElt, ModMatRngElt
Alternative Models
LegendreModel(C) : CrvCon -> CrvCon, MapIsoSch
ReducedLegendreModel(C) : CrvCon -> CrvCon, MapIsoSch
Local Conditions for Conics
BadPrimes(C) : CrvCon -> SeqEnum
Norm Residue Symbol
NormResidueSymbol(a,b,p) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
Example CrvCon_LocalGlobal (H86E4)
Solubility Certificates
HasSolubilityCertificate(C) : CrvCon -> BoolElt, SeqEnum
HasSolubilityCertificate(S) : SeqEnum[RngIntElt] -> BoolElt, SeqEnum
SolubilityCertificate(C) : CrvCon -> SeqEnum
Example CrvCon_ConicAccess (H86E5)
Finding Points
HasRationalPoint(C) : CrvCon -> BoolElt, Pt
RationalPoint(C) : CrvCon -> Pt
Random(C) : CrvCon -> Pt
RationalPoints(C) : CrvCon -> SetIndx
Example CrvCon_rational-point-enum (H86E6)
Point Reduction
IsReduced(p) : Pt -> BoolElt
Reduction(p: parameters) : Pt -> Pt
Example CrvCon_PointReduction (H86E7)
Example CrvCon_PointFinding (H86E8)
Isomorphisms with Standard Models
Conic(C) : Crv -> MapSch
Example CrvCon_ (H86E9)
ParametrizationMatrix(C) : CrvCon -> ModMatRngElt
Example CrvCon_ (H86E10)
Parametrization(P,C) : Prj, Crv -> MapSch
Isomorphism(C,D) : Crv, Crv -> MapIsoSch
Isomorphism(X,C,p) : Sch, Crv, Pt -> MapIsoSch
Example CrvCon_RationalParametrization (H86E11)
Isomorphisms of Rational Curves
Isomorphism(C,D,S,T) : CrvRat, CrvRat, [Pt], [Pt] -> MapIsoSch
Isomorphisms of Conics
Isomorphism(C,D) : CrvCon, CrvCon -> MapIsoSch
IsIsomorphic(C,D) : CrvCon, CrvCon -> BoolElt, MapIsoSch
Automorphisms of Rational Curves
Automorphism(C,S,T) : CrvRat, SetIndx, SetIndx -> MapIsoSch
Automorphisms of Conics
QuaternionAlgebra(C) : CrvCon-> AlgQuat
Automorphism(C,a) : CrvCon, AlgQuatElt -> MapIsoSch