Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: i[0](p) / i[1](p)x/yA ValueError is raised if self has no rational pointsage: C = Conic(x^2 + 2*y^2 + z^2)sage: C.parametrization()Traceback (most recent call last):...ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 haA ValueError is raised if self is not smoothsage: C = Conic(x^2 + y^2)sage: C.parametrization()Traceback (most recent call last):...ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^236 Chapter 7. Projective plane conics over Q
CHAPTEREIGHTPROJECTIVE PLANE CONICS OVERFINITE FIELDSAUTHORS:• Marco Streng (2010-07-20)class sage.schemes.plane_conics.con_finite_field.ProjectiveConic_finite_field(A,f )Bases:sage.schemes.plane_conics.con_field.ProjectiveConic_field,sage.schemes.plane_curves.projective_curve.ProjectiveCurve_finite_fieldCreate a projective plane conic curve over a finite field. See Conic for full documentation.EXAMPLES:sage: K. = FiniteField(9, ’a’)sage: P. = K[]sage: Conic(X^2 + Y^2 - a*Z^2)Projective Conic Curve over Finite Field in a of size 3^2 defined by X^2 + Y^2 + (-a)*Z^2TESTS:sage: K. = FiniteField(4, ’a’)sage: Conic([a, 1, -1])._test_pickling()count_points(n)If the base field B of self is finite of order q, then returns the number of points over F q , ..., F q n.EXAMPLES:sage: P. = GF(3)[]sage: c = Curve(x^2+y^2+z^2); cProjective Conic Curve over Finite Field of size 3 defined by x^2 + y^2 + z^2sage: c.count_points(4)[4, 10, 28, 82]has_rational_point(point=False, read_cache=True, algorithm=’default’)Always returns True because self has a point defined over its finite base field B.If point is True, then returns a second output S, which is a rational point if one exists.Points are cached. If read_cache is True, then cached information is used for the output if available.If no cached point is available or read_cache is False, then random y-coordinates are tried if self issmooth and a singular point is returned otherwise.EXAMPLES:37
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Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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