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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
Given a conic C, returns the discriminant of C. The discriminant
of a curve with defining equation
a1x2 + a2xy + a3xz
+ a2y2 + a3yz + a3z2 = 0
is defined to be the value of the degree 3 form
4a1a2a3 - a1a32
- a22a3 + a2a3a3 - a32a2.
Over any ring in which 2 is invertible, this is just 1/2
times the determinant of the matrix
pmatrix(
2a1 & a2 & a3 cr
a2 & 2a2 & a3 cr
a3 & a3 & 2a3).
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.
The Legendre polynomial of the conic C, a diagonalized defining
polynomial ax2 + by2 + cz2 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.
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.
Returns the Legendre model of the conic C --- an isomorphic curve
of the form
ax2 + by2 + cz2 = 0,
together with an isomorphism to this model.
The reduced Legendre model of the conic C over Q, that is, a
curve in the diagonal form ax2 + by2 + cz2 = 0 whose coefficients
are pairwise coprime and square-free. As a second value, the
isomorphism from C to this model is returned.
Return a conic, the matrix of whose defining polynomial has smaller
discriminant than that of the conic C. The algorithm used is the minimization part
of Simon's algorithm, ([Sim05]), used in HasRationalPoint.
A map from the conic returned to C is also returned.
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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