Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1sage: E = EllipticCurve(’101a’)sage: C = Curve(E)sage: C.plot()sage: C.plot(patch=0)sage: C.plot(patch=1)A hyperelliptic curve:sage: P. = QQ[]sage: f = 4*x^5 - 30*x^3 + 45*x - 22sage: C = HyperellipticCurve(f)sage: C.plot()sage: C.plot(patch=0)sage: C.plot(patch=1)class sage.schemes.plane_curves.projective_curve.ProjectiveCurve_prime_finite_field(A,f )Bases: sage.schemes.plane_curves.projective_curve.ProjectiveCurve_finite_fieldrational_points(algorithm=’enum’, sort=True)INPUT:•algorithm - string:•’enum’ - straightforward enumeration•’bn’ - via Singular’s brnoeth package.EXAMPLE:sage: x, y, z = PolynomialRing(GF(5), 3, ’xyz’).gens()sage: f = y^2*z^7 - x^9 - x*z^8sage: C = Curve(f); CProjective Curve over Finite Field of size 5 defined by-x^9 + y^2*z^7 - x*z^8sage: C.rational_points()[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),(3 : 1 : 1), (3 : 4 : 1)]sage: C = Curve(x - y + z)sage: C.rational_points()[(0 : 1 : 1), (1 : 1 : 0), (1 : 2 : 1), (2 : 3 : 1),(3 : 4 : 1), (4 : 0 : 1)]sage: C = Curve(x*z+z^2)sage: C.rational_points(’all’)[(0 : 1 : 0), (1 : 0 : 0), (1 : 1 : 0), (2 : 1 : 0),(3 : 1 : 0), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1),(4 : 2 : 1), (4 : 3 : 1), (4 : 4 : 1)]Note: The Brill-Noether package does not always work (i.e., the ‘bn’ algorithm. When it fails a RuntimeErrorexception is raised.riemann_roch_basis(D)Return a basis for the Riemann-Roch space corresponding to D.This uses Singular’s Brill-Noether implementation.INPUT:•D - a divisor12 Chapter 3. Projective plane curves over a general ring
Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1OUTPUT:A list of function field elements that form a basis of the Riemann-Roch spaceEXAMPLE:sage: R. = GF(2)[]sage: f = x^3*y + y^3*z + x*z^3sage: C = Curve(f); pts = C.rational_points()sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])sage: C.riemann_roch_basis(D)[x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]sage: R. = GF(5)[]sage: f = x^7 + y^7 + z^7sage: C = Curve(f); pts = C.rational_points()sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])sage: C.riemann_roch_basis(D)[(-2*x + y)/(x + y), (-x + z)/(x + y)]Note: Currently this only works over prime field and divisors supported on rational points.class sage.schemes.plane_curves.projective_curve.ProjectiveSpaceCurve_generic(A,X)Bases: sage.schemes.plane_curves.curve.Curve_generic_projective13
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BIBLIOGRAPHY[WpJacobianVariety] htt
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Sage Reference Manual: Elliptic and Plane Curves - Mirrors

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