Introduction

 
Rational Curves and Conics

      Rational Curve and Conic Creation
            Conic(coeffs) : [RngElt] -> CrvCon
            Conic(M) : Mtrx -> CrvCon
            Conic(X, f) : Prj, RngMPolElt -> CrvCon
            IsConic(S) : Sch -> BoolElt, CrvCon
            RationalCurve(X, f) : Prj, RngMPolElt -> CrvRat
            IsRationalCurve(S) : Sch -> BoolElt, CrvRat
            Example CrvCon_ConicCreation (H130E1)

      Access Functions
            DefiningPolynomial(C) : Crv -> RngMPolElt
            DefiningIdeal(C) : Crv -> RngMPol

      Rational Curve and Conic Examples
            Example CrvCon_RationalCurveExample (H130E2)
            Example CrvCon_ConicCurve (H130E3)
            Example CrvCon_ConicAccess (H130E4)

 
Conics

      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

      Other Functions on Conics
            MinimalModel(C) : CrvCon -> CrvCon, Map
            Example CrvCon_ConicMinimalModel (H130E5)

 
Local-Global Correspondence

      Local Conditions for Conics
            BadPrimes(C) : CrvCon -> SeqEnum

      Local Solubility
            IsLocallySolvable(C) : CrvCon -> BoolElt

      Norm Residue Symbol
            NormResidueSymbol(a, b, p) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
            HilbertSymbol(a, b, p : parameters) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
            Example CrvCon_LocalGlobal (H130E6)

 
Rational Points on Conics

      Finding Points
            HasRationalPoint(C) : CrvCon -> BoolElt, Pt
            RationalPoint(C) : CrvCon -> Pt
            Random(C : parameters) : CrvCon -> Pt
            Points(C : parameters) : CrvCon -> SetIndx
            Example CrvCon_rational-point-enum (H130E7)

      Point Reduction
            IsReduced(p) : Pt -> BoolElt
            Reduction(p) : Pt -> Pt
            Example CrvCon_PointReduction (H130E8)
            Example CrvCon_PointFinding (H130E9)

 
Isomorphisms

      Isomorphisms with Standard Models
            Conic(C) : Crv -> MapSch
            Example CrvCon_ (H130E10)
            ParametrizationMatrix(C) : CrvCon -> ModMatRngElt
            Example CrvCon_ (H130E11)
            Parametrization(C) : CrvCon -> MapSch
            ParametrizeOrdinaryCurve(C) : Crv -> MapSch
            Example CrvCon_RationalParametrization (H130E12)

 
Automorphisms

      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
            Example CrvCon_ConicAutomorphisms (H130E13)

 
Bibliography

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