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.
Given a conic C, returns the discriminant of C. The discriminant of a conic with defining equationa11x2 + a12xy + a13xz + a22y2 + a23yz + a33z2 = 0
is defined to be the value of the degree 3 form
4a11a22a33 - a11a232 - a122a33 + a12a13a23 - a132a22.
Over any ring in which 2 is invertible this is just 1/2 times the determinant of the matrix
pmatrix( 2a11 & a12 & a13
a12 & 2a22 & a23
a13 & a23 & 2a33).
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.
Returns the Legendre polynomial of the conic C, a diagonalised defining polynomial of the form ax2 + by2 + cz2. 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.
Returns the reduced Legendre polynomial of the conic C, which must be defined over Q or 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.
Returns the Legendre model of the conic C --- an isomorphic curve of the formax2 + by2 + cz2 = 0,
together with an isomorphism to this model.
Returns the reduced Legendre model of the conic C, which must be defined over Q or Z; that is, a curve in the diagonal form ax2 + by2 + cz2 = 0 whose coefficients are pairwise coprime and square-free. The isomorphism from C to this model is returned as a second value.
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 [Sim05], as used in HasRationalPoint. A map from the conic to C is also returned.
> 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