Coverage for local/lib/python2.7/site-packages/sage/modular/abvar/torsion_point.py : 63%
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""" Torsion points on modular abelian varieties
AUTHORS:
- William Stein (2007-03)
- Peter Bruin (2014-12): move TorsionPoint to a separate file """ # **************************************************************************** # Copyright (C) 2007 William Stein <wstein@gmail.com> # Copyright (C) 2014 Peter Bruin <P.J.Bruin@math.leidenuniv.nl> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ # ****************************************************************************
from sage.structure.element import ModuleElement from sage.structure.richcmp import richcmp, rich_to_bool
class TorsionPoint(ModuleElement): r""" An element of a finite subgroup of a modular abelian variety.
INPUT:
- ``parent`` -- a finite subgroup of a modular abelian variety
- ``element`` -- a `\QQ`-vector space element that represents this element in terms of the ambient rational homology
- ``check`` -- bool (default: ``True``): whether to check that element is in the appropriate vector space
EXAMPLES:
The following calls the :class:`TorsionPoint` constructor implicitly::
sage: J = J0(11) sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1 sage: type(G.0) <class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice_with_category.element_class'> """ def __init__(self, parent, element, check=True): """ Initialize ``self``.
EXAMPLES::
sage: J = J0(11) sage: G = J.finite_subgroup([[1/2,0], [0,1/2]]) sage: TestSuite(G).run() # long time """ raise TypeError("element must be a vector in the abelian variety's rational homology (embedded in the ambient Jacobian product)")
def element(self): r""" Return a vector over `\QQ` defining ``self``.
OUTPUT:
- A vector in the rational homology of the ambient modular Jacobian variety.
EXAMPLES:
We create some elements of `J_0(11)`::
sage: J = J0(11) sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1 sage: G.0.element() (1/3, 0)
The underlying element is a vector over the rational numbers::
sage: v = (G.0-G.1).element(); v (1/3, -1/5) sage: type(v) <type 'sage.modules.vector_rational_dense.Vector_rational_dense'> """
def _repr_(self): r""" Return a string representation of ``self``.
.. note::
Since they are represented as equivalences classes of rational homology modulo integral homology, we represent an element corresponding to `v` in the rational homology by ``[v]``.
EXAMPLES::
sage: J = J0(11) sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1 sage: G.0._repr_() '[(1/3, 0)]' """
def _add_(self, other): """ Add two finite subgroup elements with the same parent. This is called implicitly by +.
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0._add_(G.1) [(1/3, 1/5)] sage: G.0 + G.1 [(1/3, 1/5)] """
def _sub_(self, other): """ Subtract two finite subgroup elements with the same parent. This is called implicitly by +.
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0._sub_(G.1) [(1/3, -1/5)] sage: G.0 - G.1 [(1/3, -1/5)] """
def _neg_(self): """ Negate a finite subgroup element.
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0._neg_() [(-1/3, 0)] """
def _rmul_(self, left): """ Left multiply a finite subgroup element by an integer.
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0._rmul_(2) [(2/3, 0)] sage: 2*G.0 [(2/3, 0)] """
def _lmul_(self, right): """ Right multiply a finite subgroup element by an integer.
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0._lmul_(2) [(2/3, 0)] sage: G.0 * 2 [(2/3, 0)] """
def _richcmp_(self, right, op): """ Compare ``self`` and ``right``.
INPUT:
- ``self, right`` -- elements of the same finite abelian variety subgroup.
- ``op`` -- comparison operator (see :mod:`sage.structure.richcmp`)
OUTPUT: boolean
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0 > G.1 True sage: G.0 == G.0 True sage: 3*G.0 == 0 True sage: 3*G.0 == 5*G.1 True
We make sure things that should not be equal are not::
sage: H = J0(14).finite_subgroup([[1/3,0]]) sage: G.0 == H.0 False sage: G.0 [(1/3, 0)] sage: H.0 [(1/3, 0)] """
def additive_order(self): """ Return the additive order of ``self``.
EXAMPLES::
sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]]) sage: G.0.additive_order() 3 sage: G.1.additive_order() 5 sage: (G.0 + G.1).additive_order() 15 sage: (3*G.0).additive_order() 1 """
def _relative_element(self): """ Return coordinates of ``self`` on a basis for the integral homology of the containing abelian variety.
OUTPUT: vector
EXAMPLES::
sage: A = J0(43)[1]; A Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: C = A.cuspidal_subgroup(); C Finite subgroup with invariants [7] over QQ of Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: x = C.0; x [(0, 1/7, 0, 6/7, 0, 5/7)] sage: x._relative_element() (0, 1/7, 6/7, 5/7) """ # check=False prevents testing that the element is really in # the lattice, not just in the corresponding QQ-vector space. |