Coverage for local/lib/python2.7/site-packages/sage/schemes/jacobians/abstract_jacobian.py : 57%
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""" Base class for Jacobians of curves """
#******************************************************************************* # Copyright (C) 2005 William Stein # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #*******************************************************************************
from sage.categories.fields import Fields _Fields = Fields() from sage.schemes.generic.scheme import Scheme, is_Scheme from sage.structure.richcmp import richcmp_method, richcmp
def is_Jacobian(J): """ Return True if `J` is of type Jacobian_generic.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian, is_Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: J = Jacobian(C) sage: is_Jacobian(J) True
::
sage: E = EllipticCurve('37a1') sage: is_Jacobian(E) False """
def Jacobian(C): """ EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: Jacobian(C) Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 """
@richcmp_method class Jacobian_generic(Scheme): """ Base class for Jacobians of projective curves.
The input must be a projective curve over a field.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: J = Jacobian(C); J Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 """ def __init__(self, C): """ TESTS::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: J = Jacobian_generic(C); J Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 sage: type(J) <class 'sage.schemes.jacobians.abstract_jacobian.Jacobian_generic_with_category'>
Note: this is an abstract parent, so we skip element tests::
sage: TestSuite(J).run(skip =["_test_an_element",\ "_test_elements",\ "_test_elements_eq_reflexive",\ "_test_elements_eq_symmetric",\ "_test_elements_eq_transitive",\ "_test_elements_neq",\ "_test_some_elements"])
::
sage: Jacobian_generic(ZZ) Traceback (most recent call last): ... TypeError: Argument (=Integer Ring) must be a scheme. sage: Jacobian_generic(P2) Traceback (most recent call last): ... ValueError: C (=Projective Space of dimension 2 over Rational Field) must have dimension 1. sage: P2.<x, y, z> = ProjectiveSpace(Zmod(6), 2) sage: C = Curve(x + y + z) sage: Jacobian_generic(C) Traceback (most recent call last): ... TypeError: C (=Projective Plane Curve over Ring of integers modulo 6 defined by x + y + z) must be defined over a field. """
def __richcmp__(self, J, op): """ Compare the Jacobian self to `J`. If `J` is a Jacobian, then self and `J` are equal if and only if their curves are equal.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: J1 = Jacobian(Curve(x^3 + y^3 + z^3)) sage: J1 == J1 True sage: J1 == P2 False sage: J1 != P2 True sage: J2 = Jacobian(Curve(x + y + z)) sage: J1 == J2 False sage: J1 != J2 True """
def _repr_(self): """ Return a string representation of this Jacobian.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: J = Jacobian(Curve(x^3 + y^3 + z^3)); J Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 sage: J._repr_() 'Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3' """
def _point(self): """ Return the Hom-set from some affine scheme to ``self``.
OUTPUT:
This method always raises a ``NotImplementedError``; it is only abstract.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: J = Jacobian(Curve(x^3 + y^3 + z^3)) sage: J._point() Traceback (most recent call last): ... NotImplementedError """
def curve(self): """ Return the curve of which self is the Jacobian.
EXAMPLES::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: J = Jacobian(Curve(x^3 + y^3 + z^3)) sage: J.curve() Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 """
def change_ring(self, R): r""" Return the Jacobian over the ring `R`.
INPUT:
- ``R`` -- a field. The new base ring.
OUTPUT:
The Jacobian over the ring `R`.
EXAMPLES::
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^3-10*x+9) sage: Jac = H.jacobian(); Jac Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^3 - 10*x + 9 sage: Jac.change_ring(RDF) Jacobian of Hyperelliptic Curve over Real Double Field defined by y^2 = x^3 - 10.0*x + 9.0 """
def base_extend(self, R): r""" Return the natural extension of ``self`` over `R`
INPUT:
- ``R`` -- a field. The new base field.
OUTPUT:
The Jacobian over the ring `R`.
EXAMPLES::
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^3-10*x+9) sage: Jac = H.jacobian(); Jac Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^3 - 10*x + 9 sage: F.<a> = QQ.extension(x^2+1) sage: Jac.base_extend(F) Jacobian of Hyperelliptic Curve over Number Field in a with defining polynomial x^2 + 1 defined by y^2 = x^3 - 10*x + 9 """ raise ValueError('Not a field: ' + str(R)) return self raise ValueError('no natural map from the base ring (=%s) to R (=%s)!' % (self.base_ring(), R)) |