Sage Reference Manual: Elliptic and Plane Curves - Mirrors

Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.141/4sage: C.symmetric_matrix().determinant()41/4Determinants are only defined in characteristic different from 2:sage: C = Conic(GF(2), [1, 1, 1, 1, 1, 0])sage: C.is_smooth()Truesage: C.determinant()Traceback (most recent call last):...ValueError: The conic self (= Projective Conic Curve over Finite Field of size 2 defined bydiagonal_matrix()Returns a diagonal matrix D and a matrix T such that T t AT = D holds, where (x, y, z)A(x, y, z) t is thedefining polynomial of the conic self.EXAMPLES:sage: c = Conic(QQ, [1,2,3,4,5,6])sage: d, t = c.diagonal_matrix(); d, t([ 1 0 0] [ 1 -1 -7/6][ 0 3 0] [ 0 1 -1/3][ 0 0 41/12], [ 0 0 1])sage: t.transpose()*c.symmetric_matrix()*t[ 1 0 0][ 0 3 0][ 0 0 41/12]Diagonal matrices are only defined in characteristic different from 2:sage: c = Conic(GF(4, ’a’), [0, 1, 1, 1, 1, 1])sage: c.is_smooth()Truesage: c.diagonal_matrix()Traceback (most recent call last):...ValueError: The conic self (= Projective Conic Curve over Finite Field in a of size 2^2 defidiagonalization(names=None)Returns a diagonal conic C, an isomorphism of schemes M : C -> self and the inverse N of M.EXAMPLES:sage: Conic(GF(5), [1,0,1,1,0,1]).diagonalization()(Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + 2*z^2,Scheme morphism:From: Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + 2*z^2To: Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + x*z + z^2Defn: Defined on coordinates by sending (x : y : z) to(x + 2*z : y : z),Scheme morphism:From: Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + x*z + z^2To: Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + 2*z^2Defn: Defined on coordinates by sending (x : y : z) to(x - 2*z : y : z))19