Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: X,Y,Z = QQ[’X,Y,Z’].gens()sage: Conic(X^2 - X*Y + Y^2 - Z^2)Projective Conic Curve over Rational Field defined by X^2 - X*Y + Y^2 - Z^2sage: x,y = GF(7)[’x,y’].gens()sage: Conic(x^2 - x + 2*y^2 - 3, ’U,V,W’)Projective Conic Curve over Finite Field of size 7 defined by U^2 + 2*V^2 - U*W - 3*W^2Conic curves given by matricessage: Conic(matrix(QQ, [[1, 2, 0], [4, 0, 0], [7, 0, 9]]), ’x,y,z’)Projective Conic Curve over Rational Field defined by x^2 + 6*x*y + 7*x*z + 9*z^2sage: x,y,z = GF(11)[’x,y,z’].gens()sage: C = Conic(x^2+y^2-2*z^2); CProjective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2sage: Conic(C.symmetric_matrix(), ’x,y,z’)Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2Conics given by coefficientssage: Conic(QQ, [1,2,3])Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + 3*z^2sage: Conic(GF(7), [1,2,3,4,5,6], ’X’)Projective Conic Curve over Finite Field of size 7 defined by X0^2 + 2*X0*X1 - 3*X1^2 + 3*X0*X2The conic through a set of pointssage: C = Conic(QQ, [[10,2],[3,4],[-7,6],[7,8],[9,10]]); CProjective Conic Curve over Rational Field defined by x^2 + 13/4*x*y - 17/4*y^2 - 35/2*x*z + 91/sage: C.rational_point()(10 : 2 : 1)sage: C.point([3,4])(3 : 4 : 1)sage: a=AffineSpace(GF(13),2)sage: Conic([a([x,x^2]) for x in range(5)])Projective Conic Curve over Finite Field of size 13 defined by x^2 - y*z16 Chapter 4. Plane conic constructor
CHAPTERFIVEPROJECTIVE PLANE CONICS OVER AFIELDAUTHORS:• Marco Streng (2010-07-20)• Nick Alexander (2008-01-08)class sage.schemes.plane_conics.con_field.ProjectiveConic_field(A, f )Bases: sage.schemes.plane_curves.projective_curve.ProjectiveCurve_genericCreate a projective plane conic curve over a field. See Conic for full documentation.EXAMPLES:sage: K = FractionField(PolynomialRing(QQ, ’t’))sage: P. = K[]sage: Conic(X^2 + Y^2 - Z^2)Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational FielTESTS:sage: K = FractionField(PolynomialRing(QQ, ’t’))sage: Conic([K(1), 1, -1])._test_pickling()base_extend(S)Returns the conic over S given by the same equation as self.EXAMPLES:sage: c = Conic([1, 1, 1]); cProjective Conic Curve over Rational Field defined by x^2 + y^2 + z^2sage: c.has_rational_point()Falsesage: d = c.base_extend(QuadraticField(-1, ’i’)); dProjective Conic Curve over Number Field in i with defining polynomial x^2 + 1 defined by x^sage: d.rational_point(algorithm = ’rnfisnorm’)(i : 1 : 0)cache_point(p)Replace the point in the cache of self by p for use by self.rational_point() andself.parametrization().EXAMPLES:17
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Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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