Sage Reference Manual: Elliptic and Plane Curves - Mirrors

Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: Conic([1,-1,0]).is_smooth()Falsesage: Conic(GF(2),[1,1,1,1,1,0]).is_smooth()Truematrix()Returns a matrix M such that (x, y, z)M(x, y, z) t is the defining equation of self.The matrix M is upper triangular if the base field has characteristic 2 and symmetric otherwise.EXAMPLES:sage: R. = QQ[]sage: C = Conic(x^2 + x*y + y^2 + z^2)sage: C.matrix()[ 1 1/2 0][1/2 1 0][ 0 0 1]sage: R. = GF(2)[]sage: C = Conic(x^2 + x*y + y^2 + x*z + z^2)sage: C.matrix()[1 1 1][0 1 0][0 0 1]parametrization(point=None, morphism=True)Return a parametrization f of self together with the inverse of f.If point is specified, then that point is used for the parametrization. Otherwise, useself.rational_point() to find a point.If morphism is True, then f is returned in the form of a Scheme morphism. Otherwise, it is a tuple ofpolynomials that gives the parametrization.EXAMPLES:An example over a finite fieldsage: c = Conic(GF(2), [1,1,1,1,1,0])sage: c.parametrization()(Scheme morphism:From: Projective Space of dimension 1 over Finite Field of size 2To: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y+ y^2 + x*z + y*zDefn: Defined on coordinates by sending (x : y) to(x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),Scheme morphism:From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y+ y^2 + x*z + y*zTo: Projective Space of dimension 1 over Finite Field of size 2Defn: Defined on coordinates by sending (x : y : z) to(y : x))An example with morphism = Falsesage: R. = QQ[]sage: C = Curve(7*x^2 + 2*y*z + z^2)sage: (p, i) = C.parametrization(morphism = False); (p, i)([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])sage: C.defining_polynomial()(p)23