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Magma
Computer • algebra
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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 (H127E1)
Access Functions
DefiningPolynomial(C) : Crv -> RngMPolElt
DefiningIdeal(C) : Crv -> RngMPol
Rational Curve and Conic Examples
Example
CrvCon_RationalCurve
Example
(H127E2)
Example
CrvCon_ConicCurve (H127E3)
Example
CrvCon_ConicAccess (H127E4)
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 (H127E5)
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 (H127E6)
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 (H127E7)
Point Reduction
IsReduced(p) : Pt -> BoolElt
Reduction(p) : Pt -> Pt
Example
CrvCon_PointReduction (H127E8)
Example
CrvCon_PointFinding (H127E9)
Isomorphisms
Isomorphisms with Standard Models
Conic(C) : Crv -> MapSch
Example
CrvCon_ (H127E10)
ParametrizationMatrix(C) : CrvCon -> ModMatRngElt
Example
CrvCon_ (H127E11)
Parametrization(C) : CrvCon -> MapSch
ParametrizeOrdinaryCurve(C) : Crv -> MapSch
Example
CrvCon_RationalParametrization (H127E12)
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 (H127E13)
Bibliography
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V2.28, 28 February 2025