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, parametrizations and isomorphisms of genus zero curves with standard models and finally the automorphism group of conics.

Subsections

• Elementary Invariants
• Alternative Defining Polynomials
• Alternative Models

Elementary Invariants

Discriminant(C) : CrvCon -> FldElt

Given a conic C, returns the discriminant of C. The discriminant of a curve 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

pmatrix( 2a_(11) & a_(12) & a_(13) cr a_(12) & 2a_(22) & a_(23) cr a_(13) & a_(23) & 2a_(33)).

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 diagonalized Legendre model for the curve, whose defining polynomial can be accessed.

LegendrePolynomial(C) : CrvCon -> RngMPolElt, ModMatRngElt

The Legendre polynomial of the conic C, a diagonalized defining polynomial ax^2 + by^2 + cz^2 for the curve which, once computed, is stored as an attribute. As a second value, the transformation matrix is returned, defining the isomorphism from C to the Legendre model.

ReducedLegendrePolynomial(C) : CrvCon -> RngMPolElt, ModMatRngElt

The reduced Legendre polynomial of the conic C over Q or Z, that is, a diagonalized integral polynomial whose coefficients are pairwise coprime and square-free. As a second value, the transformation matrix to this model is returned, defining the isomorphism from C to the Legendre model.

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

The reduced Legendre model of the conic C over Q, that is, a curve in the diagonal form ax^2 + by^2 + cz^2 = 0 whose coefficients are pairwise coprime and square-free. As a second value, the isomorphism from C to this model is returned.