Conics

In this section we discuss the creation and basic attributes of conics, particularly the standard models for them. In subsequent sections we treat the local-global theory and existence of points on conics, the efficient algorithms for finding rational points, parametrisations and isomorphisms of genus zero curves with standard models, and finally the automorphism group of conics.

Elementary Invariants

Discriminant(C): CrvCon -> FldElt

Given a conic \(C\), returns the discriminant of \(C\). The discriminant of a conic with defining equation

\[a_{11}x^2 + a_{12}xy + a_{13}xz + a_{22}y^2 + a_{23}yz + a_{33}z^2 = 0\]

is defined to be the value of the degree 3 form

\[4a_{11}a_{22}a_{33} - a_{11}a_{23}^2 - a_{12}^2a_{33} + a_{12}a_{13}a_{23} - a_{13}^2a_{22}.\]

Over any ring in which \(2\) is invertible this is just \(1/2\) times the determinant of the matrix

\[\begin{split}\begin{pmatrix}2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23} \\ a_{13} & a_{23} & 2a_{33}\end{pmatrix}.\end{split}\]

Alternative Defining Polynomials

The functions described here provide access to basic information stored for a conic \(C\). In addition to the defining polynomial, curves over the rationals compute and store a diagonalised Legendre model for the curve, whose defining polynomial can be accessed.

LegendrePolynomial(C): CrvCon -> RngMPolElt, ModMatRngElt

Returns the Legendre polynomial of the conic \(C\), a diagonalised defining polynomial of the form \(ax^2 + by^2 + cz^2\). Once computed, this polynomial is stored as an attribute. The transformation matrix defining the isomorphism from \(C\) to the Legendre model is returned as the second value.

ReducedLegendrePolynomial(C): CrvCon -> RngMPolElt, ModMatRngElt

Returns the reduced Legendre polynomial of the conic \(C\), which must be defined over \({\mathbb{Q}}\) or \({\mathbb{Z}}\); that is, a diagonalised integral polynomial whose coefficients are pairwise coprime and square-free. The transformation matrix defining the isomorphism from \(C\) to this reduced Legendre model is returned as the second value.

Alternative Models

LegendreModel(C): CrvCon -> CrvCon, MapIsoSch

Returns the Legendre model of the conic \(C\) — an isomorphic curve of the form

\[ax^2 + by^2 + cz^2 = 0,\]

together with an isomorphism to this model.

ReducedLegendreModel(C): CrvCon -> CrvCon, MapIsoSch

Returns the reduced Legendre model of the conic \(C\), which must be defined over \({\mathbb{Q}}\) or \({\mathbb{Z}}\); that is, a curve in the diagonal form \(ax^2+by^2+cz^2 = 0\) whose coefficients are pairwise coprime and square-free. The isomorphism from \(C\) to this model is returned as a second value.

Other Functions on Conics

MinimalModel(C): CrvCon -> CrvCon, Map

Returns a conic, the matrix of whose defining polynomial has smaller discriminant than that of the conic \(C\) (where possible). The algorithm used is the minimisation stage of Simon’s algorithm [Simon, 2005], as used in HasRationalPoint. A map from the conic to \(C\) is also returned.

Example: Conic Minimal Model (ex-4f68ee)

In the following example we are able to reduce the conic at 13.

> P2<x,y,z> := ProjectiveSpace(RationalField(), 2);
> f := 123*x^2 + 974*x*y - 417*x*z + 654*y^2 + 113*y*z - 65*z^2;
> C := Conic(P2, f);
> BadPrimes(C);
[ 491, 18869 ]
> [ x[1] : x in Factorization(Integers()!Discriminant(C)) ];
[ 13, 491, 18869 ]
> MinimalModel(C);
Conic over Rational Field defined by
-9*x^2 + 4*x*y + 6*x*z + 564*y^2 + 178*y*z + 1837*z^2
Mapping from: Conic over Rational Field defined by
-9*x^2 + 4*x*y + 6*x*z + 564*y^2 + 178*y*z + 1837*z^2 to CrvCon: C
with equations :
x + 6*y - 10*z
-x - 8*y + 4*z
-8*x - 50*y + 61*z
and inverse
-144/13*x + 67/13*y - 28/13*z
29/26*x - 19/26*y + 3/13*z
-7/13*x + 1/13*y - 1/13*z

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