finite_subgroup.py

r"""
Finite subgroups of modular abelian varieties

Sage can compute with fairly general finite subgroups of modular
abelian varieties. Elements of finite order are represented by
equivalence classes of elements in `H_1(A,\QQ)`
modulo `H_1(A,\ZZ)`. A finite subgroup can be
defined by giving generators and via various other constructions.
Given a finite subgroup, one can compute generators, as well as the
structure as an abstract group. Arithmetic on subgroups is also
supported, including adding two subgroups together, checking
inclusion, etc.

TODO: Intersection, action of Hecke operators.

AUTHORS:

- William Stein (2007-03)

EXAMPLES::

    sage: J = J0(33)
    sage: C = J.cuspidal_subgroup()
    sage: C
    Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
    sage: C.order()
    100
    sage: C.gens()
    [[(1/10, 0, 1/10, 1/10, 1/10, 3/10)], [(0, 1/5, 1/10, 0, 1/10, 9/10)], [(0, 0, 1/2, 0, 1/2, 1/2)]]
    sage: C.0 + C.1
    [(1/10, 1/5, 1/5, 1/10, 1/5, 6/5)]
    sage: 10*(C.0 + C.1)
    [(0, 0, 0, 0, 0, 0)]
    sage: G = C.subgroup([C.0 + C.1]); G
    Finite subgroup with invariants [10] over QQbar of Abelian variety J0(33) of dimension 3
    sage: G.gens()
    [[(1/10, 1/5, 1/5, 1/10, 1/5, 1/5)]]
    sage: G.order()
    10
    sage: G <= C
    True
    sage: G >= C
    False

We make a table of the order of the cuspidal subgroup for the first
few levels::

    sage: for N in range(11,40): print N, J0(N).cuspidal_subgroup().order()
    ...
    11 5
    12 1
    13 1
    14 6
    15 8
    16 1
    17 4
    18 1
    19 3
    20 6
    21 8
    22 25
    23 11
    24 8
    25 1
    26 21
    27 9
    28 36
    29 7
    30 192
    31 5
    32 8
    33 100
    34 48
    35 48
    36 12
    37 3
    38 135
    39 56

TESTS::

    sage: G = J0(11).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: loads(dumps(G)) == G
    True
    sage: loads(dumps(G.0)) == G.0
    True
"""

###########################################################################
#       Copyright (C) 2007 William Stein <[email protected]>               #
#  Distributed under the terms of the GNU General Public License (GPL)    #
#                  http://www.gnu.org/licenses/                           #
###########################################################################

from sage.modules.module      import Module_old
from sage.modules.free_module import is_FreeModule
from sage.structure.element   import ModuleElement
from sage.structure.sequence  import Sequence
from sage.rings.all           import gcd, lcm, QQ, ZZ, QQbar, is_Field, Integer, composite_field
from sage.misc.misc           import prod

import abvar as abelian_variety

class FiniteSubgroup(Module_old):
    def __init__(self, abvar, field_of_definition=QQ):
        """
        A finite subgroup of a modular abelian variety.
        
        INPUT:
        
        
        -  ``abvar`` - a modular abelian variety
        
        -  ``field_of_definition`` - a field over which this
           group is defined.
        
        
        EXAMPLES: This is an abstract base class, so there are no instances
        of this class itself.
        
        ::
        
            sage: A = J0(37)
            sage: G = A.torsion_subgroup(3); G
            Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
            sage: type(G)
            <class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice'>
            sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup
            sage: isinstance(G, FiniteSubgroup)
            True
        """
        if not is_Field(field_of_definition):
            raise TypeError, "field_of_definition must be a field"
        if not abelian_variety.is_ModularAbelianVariety(abvar):
            raise TypeError, "abvar must be a modular abelian variety"
        Module_old.__init__(self, ZZ)
        self.__abvar = abvar
        self.__field_of_definition = field_of_definition

    ################################################################
    # DERIVED CLASS MUST OVERRIDE THE lattice METHOD
    ################################################################    
    def lattice(self):
        """
        Return the lattice corresponding to this subgroup in the rational
        homology of the modular Jacobian product. The elements of the
        subgroup are represented by vectors in the ambient vector space
        (the rational homology), and this returns the lattice they span.
        EXAMPLES::
        
            sage: J = J0(33); C = J[0].cuspidal_subgroup(); C
            Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
            sage: C.lattice()
            Free module of degree 6 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1/5 13/5   -2 -4/5    2 -1/5]
            [   0    3   -2   -1    2    0]
        """
        raise NotImplementedError

    def _relative_basis_matrix(self):
        """
        Return matrix of this finite subgroup, but relative to the homology
        of the parent abelian variety.
        
        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: C._relative_basis_matrix()
            [  1   0   0   0]
            [  0 1/7 6/7 5/7]
            [  0   0   1   0]
            [  0   0   0   1]
        """
        try:
            return self.__relative_basis_matrix
        except AttributeError:
            M = self.__abvar.lattice().coordinate_module(self.lattice()).basis_matrix()
            self.__relative_basis_matrix = M
            return M

    # General functionality
    def __cmp__(self, other):
        """
        Compare this finite subgroup to other.
        
        If other is not a modular abelian variety finite subgroup, then the
        types of self and other are compared. If other is a finite
        subgroup, and the ambient abelian varieties are equal, then the
        subgroups themselves are compared, by comparing their full modules.
        If the containing abelian varieties are not equal and their ambient
        varieties are different they are compared; if they are the same,
        then a NotImplemnetedError is raised (this is temporary).
        
        EXAMPLES: We first compare to subgroups of `J_0(37)`::
        
            sage: A = J0(37)
            sage: G = A.torsion_subgroup(3); G.order()
            81
            sage: H = A.cuspidal_subgroup(); H.order()
            3
            sage: H < G
            True
            sage: H.is_subgroup(G)
            True
            sage: H < 5 #random (meaningless since it depends on memory layout)
            False
            sage: 5 < H #random (meaningless since it depends on memory layout)
            True
        
        The ambient varieties are compared::
        
            sage: cmp(A[0].cuspidal_subgroup(), J0(11).cuspidal_subgroup())
            1
        
        Comparing subgroups sitting in different abelian varieties::
        
            sage: cmp(A[0].cuspidal_subgroup(), A[1].cuspidal_subgroup())
            -1
        """
        if not isinstance(other, FiniteSubgroup):
            return cmp(type(self), type(other))
        A = self.abelian_variety()
        B = other.abelian_variety()
        if not A.in_same_ambient_variety(B):
            return cmp(A.ambient_variety(), B.ambient_variety())
        L = A.lattice() + B.lattice()
        # Minus sign because order gets reversed in passing to lattices.
        return -cmp(self.lattice() + L, other.lattice() + L)

    def is_subgroup(self, other):
        """
        Return True exactly if self is a subgroup of other, and both are
        defined as subgroups of the same ambient abelian variety.
        
        EXAMPLES::
        
            sage: C = J0(22).cuspidal_subgroup()
            sage: H = C.subgroup([C.0])
            sage: K = C.subgroup([C.1])
            sage: H.is_subgroup(K)
            False
            sage: K.is_subgroup(H)
            False
            sage: K.is_subgroup(C)
            True
            sage: H.is_subgroup(C)
            True
        """
        # We use that self is contained in other, whether other is
        # either a finite group or an abelian variety, if and only
        # if self doesn't shrink when intersected with other. 
        try:
            return self.intersection(other).order() == self.order()
        except TypeError:
            return False

    def __add__(self, other):
        """
        Return the sum of two subgroups.
        
        EXAMPLES::
        
            sage: C = J0(22).cuspidal_subgroup()
            sage: C.gens()
            [[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
            sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
            sage: A + B == C
            True
        """
        if not isinstance(other, FiniteSubgroup):
            raise TypeError, "only addition of two finite subgroups is defined"
        A = self.abelian_variety()
        B = other.abelian_variety()
        if not A.in_same_ambient_variety(B):
            raise ValueError("self and other must be in the same ambient Jacobian")
        K = composite_field(self.field_of_definition(), other.field_of_definition())
        lattice = self.lattice() + other.lattice()
        if A != B:
            lattice += C.lattice()
        
        return FiniteSubgroup_lattice(self.abelian_variety(), lattice, field_of_definition=K)

    def exponent(self):
        """
        Return the exponent of this finite abelian group.
        
        OUTPUT: Integer
        
        EXAMPLES::
        
            sage: t = J0(33).hecke_operator(7)
            sage: G = t.kernel()[0]; G
            Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
            sage: G.exponent()
            4
        """
        try:
            return self.__exponent
        except AttributeError:
            e = lcm(self.invariants())
            self.__exponent = e
            return e

    def intersection(self, other):
        """
        Return the intersection of the finite subgroups self and other.
        
        INPUT:
        
        
        -  ``other`` - a finite group
        
        
        OUTPUT: a finite group
        
        EXAMPLES::
        
            sage: E11a0, E11a1, B = J0(33)
            sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
            sage: G.intersection(H)
            Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
            sage: W = E11a1.torsion_subgroup(15)
            sage: G.intersection(W)
            Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
            sage: E11a0.intersection(E11a1)[0]
            Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
        
        We intersect subgroups of different abelian varieties.
        
        ::
        
            sage: E11a0, E11a1, B = J0(33)
            sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
            sage: G.intersection(H)
            Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
            sage: E11a0.intersection(E11a1)[0]
            Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
        
        We intersect abelian varieties with subgroups::
        
            sage: t = J0(33).hecke_operator(7)
            sage: G = t.kernel()[0]; G
            Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
            sage: A = J0(33).old_subvariety()
            sage: A.intersection(G)
            Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
            sage: A.hecke_operator(7).kernel()[0]
            Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
            sage: B = J0(33).new_subvariety()
            sage: B.intersection(G)
            Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
            sage: B.hecke_operator(7).kernel()[0]
            Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
            sage: A.intersection(B)[0]
            Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
        """
        A = self.abelian_variety()
        if abelian_variety.is_ModularAbelianVariety(other):
            amb = other
            B = other
            M = B.lattice().scale(Integer(1)/self.exponent())
            K = composite_field(self.field_of_definition(), other.base_field())
        else:
            amb = A
            if not isinstance(other, FiniteSubgroup):
                raise TypeError, "only addition of two finite subgroups is defined"
            B = other.abelian_variety()
            if A.ambient_variety() != B.ambient_variety():
                raise TypeError, "finite subgroups must be in the same ambient product Jacobian"
            M = other.lattice()
            K = composite_field(self.field_of_definition(), other.field_of_definition())

        L = self.lattice()
        if A != B:
            # TODO: This might be way slower than what we could do if
            # we think more carefully.
            C = A + B
            L = L + C.lattice()
            M = M + C.lattice()
        W = L.intersection(M).intersection(amb.vector_space())
        return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
        
    def __mul__(self, right):
        """
        Multiply this subgroup by the rational number right.
        
        If right is an integer the result is a subgroup of self. If right
        is a rational number `n/m`, then this group is first
        divided by `m` then multiplied by `n`.
        
        INPUT:
        
        
        -  ``right`` - a rational number
        
        
        OUTPUT: a subgroup
        
        EXAMPLES::
        
            sage: J = J0(37)
            sage: H = J.cuspidal_subgroup(); H.order()
            3
            sage: G = H * 3; G.order()
            1
            sage: G = H * (1/2); G.order()
            48
            sage: J.torsion_subgroup(2) + H == G
            True
            sage: G = H*(3/2); G.order()
            16        
            sage: J = J0(42)
            sage: G = J.cuspidal_subgroup(); factor(G.order())
            2^8 * 3^2
            sage: (G * 3).order()
            256
            sage: (G * 0).order()
            1
            sage: (G * (1/5)).order()
            22500000000
        """
        lattice = self.lattice().scale(right)
        return FiniteSubgroup_lattice(self.abelian_variety(), lattice,
                                      field_of_definition = self.field_of_definition())

    def __rmul__(self, left):
        """
        Multiply this finite subgroup on the left by an integer.
        
        EXAMPLES::
        
            sage: J = J0(42)
            sage: G = J.cuspidal_subgroup(); factor(G.order())
            2^8 * 3^2
            sage: H = G.__rmul__(2)
            sage: H.order().factor()
            2^4 * 3^2
            sage: 2*G 
            Finite subgroup with invariants [6, 24] over QQ of Abelian variety J0(42) of dimension 5
        """
        return self * left

    def abelian_variety(self):
        """
        Return the abelian variety that this is a finite subgroup of.
        
        EXAMPLES::
        
            sage: J = J0(42)
            sage: G = J.rational_torsion_subgroup(); G
            Torsion subgroup of Abelian variety J0(42) of dimension 5
            sage: G.abelian_variety()
            Abelian variety J0(42) of dimension 5
        """
        return self.__abvar

    def field_of_definition(self):
        """
        Return the field over which this finite modular abelian variety
        subgroup is defined. This is a field over which this subgroup is
        defined.
        
        EXAMPLES::
        
            sage: J = J0(42)
            sage: G = J.rational_torsion_subgroup(); G
            Torsion subgroup of Abelian variety J0(42) of dimension 5
            sage: G.field_of_definition()
            Rational Field
        """
        return self.__field_of_definition

    def _repr_(self):
        """
        Return string representation of this finite subgroup.
        
        EXAMPLES::
        
            sage: J = J0(42)
            sage: G = J.torsion_subgroup(3); G._repr_()
            'Finite subgroup with invariants [3, 3, 3, 3, 3, 3, 3, 3, 3, 3] over QQ of Abelian variety J0(42) of dimension 5'
        """
        K = self.__field_of_definition
        if K == QQbar:
            field = "QQbar"
        elif K == QQ:
            field = "QQ"
        else:
            field = str(K)
        return "Finite subgroup %sover %s of %s"%(self._invariants_repr(), field, self.__abvar)

    def _invariants_repr(self):
        """
        The string representation of the 'invariants' part of this group.
        
        We make this a separate function so it is possible to create finite
        subgroups that don't print their invariants, since printing them
        could be expensive.
        
        EXAMPLES::
        
            sage: J0(42).cuspidal_subgroup()._invariants_repr()
            'with invariants [2, 2, 12, 48] '
        """
        return 'with invariants %s '%(self.invariants(), )

    def order(self):
        """
        Return the order (number of elements) of this finite subgroup.
        
        EXAMPLES::
        
            sage: J = J0(42)
            sage: C = J.cuspidal_subgroup()
            sage: C.order()
            2304
        """
        try:
            return self.__order
        except AttributeError:
            if self.__abvar.dimension() == 0:
                self.__order = ZZ(1)
                return self.__order
            o = prod(self.invariants())
            self.__order = o
            return o

    def gens(self):
        """
        Return generators for this finite subgroup.
        
        EXAMPLES: We list generators for several cuspidal subgroups::
        
            sage: J0(11).cuspidal_subgroup().gens()
            [[(0, 1/5)]]
            sage: J0(37).cuspidal_subgroup().gens()
            [[(0, 0, 0, 1/3)]]
            sage: J0(43).cuspidal_subgroup().gens()
            [[(0, 1/7, 0, 6/7, 0, 5/7)]]
            sage: J1(13).cuspidal_subgroup().gens()
            [[(1/19, 0, 0, 9/19)], [(0, 1/19, 1/19, 18/19)]]
            sage: J0(22).torsion_subgroup(6).gens()
            [[(1/6, 0, 0, 0)], [(0, 1/6, 0, 0)], [(0, 0, 1/6, 0)], [(0, 0, 0, 1/6)]]
        """
        try:
            return self.__gens
        except AttributeError:
            pass
        
        B = [TorsionPoint(self, v) for v in self.lattice().basis() if v.denominator() > 1]
        self.__gens = Sequence(B, immutable=True)
        return self.__gens

    def gen(self, n):
        r"""
        Return `n^{th}` generator of self.
        
        EXAMPLES::
        
            sage: J = J0(23)
            sage: C = J.torsion_subgroup(3)
            sage: C.gens()
            [[(1/3, 0, 0, 0)], [(0, 1/3, 0, 0)], [(0, 0, 1/3, 0)], [(0, 0, 0, 1/3)]]
            sage: C.gen(0)
            [(1/3, 0, 0, 0)]
            sage: C.gen(3)
            [(0, 0, 0, 1/3)]
            sage: C.gen(4)
            Traceback (most recent call last):
            ...
            IndexError: list index out of range
        
        Negative indices wrap around::
        
            sage: C.gen(-1)
            [(0, 0, 0, 1/3)]
        """
        return self.gens()[n]
        
    def __call__(self, x):
        r"""
        Coerce `x` into this finite subgroup.
        
        This works when the abelian varieties that contains x and self are
        the same, or if `x` is coercible into the rational homology
        (viewed as an abstract `\QQ`-vector space).
        
        EXAMPLES: We first construct the `11`-torsion subgroup of
        `J_0(23)`::
        
            sage: J = J0(23)
            sage: G = J.torsion_subgroup(11)
            sage: G.invariants()
            [11, 11, 11, 11]
        
        We also construct the cuspidal subgroup.
        
        ::
        
            sage: C = J.cuspidal_subgroup()
            sage: C.invariants()
            [11]
        
        Coercing something into its parent returns it::
        
            sage: G(G.0) is G.0
            True
        
        We coerce an element from the cuspidal subgroup into the
        `11`-torsion subgroup::
        
            sage: z = G(C.0); z
            [(1/11, 10/11, 0, 8/11)]
            sage: z.parent() == G
            True
        
        We coerce a list, which defines an element of the underlying
        ``full_module`` into `G`, and verify an
        equality::
        
            sage: x = G([1/11, 1/11, 0, -1/11])
            sage: x == G([1/11, 1/11, 0, 10/11])
            True
        
        Finally we attempt to coerce in an element that shouldn't work,
        since is is not in `G`::
        
            sage: G(J.torsion_subgroup(3).0)
            Traceback (most recent call last):
            ...
            TypeError: x does not define an element of self
        """
        if isinstance(x, TorsionPoint):
            if x.parent() is self:
                return x
            elif x.parent() == self:
                return TorsionPoint(self, x.element(), check=False)
            elif x.parent().abelian_variety() == self.abelian_variety():
                return self(x.element())
            else:
                raise TypeError, "x does not define an element of self"
        else:
            z = self.abelian_variety().vector_space()(x)
            if not z in self.lattice():
                raise TypeError, "x does not define an element of self"
            return TorsionPoint(self, z, check=False)


    def __contains__(self, x):
        """
        Returns True if x is contained in this finite subgroup.
        
        EXAMPLES: We define two distinct finite subgroups of
        `J_0(27)`::
        
            sage: G1 = J0(27).rational_cusp_subgroup(); G1
            Finite subgroup with invariants [3] over QQ of Abelian variety J0(27) of dimension 1
            sage: G1.0
            [(1/3, 0)]
            sage: G2 = J0(27).cuspidal_subgroup(); G2
            Finite subgroup with invariants [3, 3] over QQ of Abelian variety J0(27) of dimension 1
            sage: G2.gens()
            [[(1/3, 0)], [(0, 1/3)]]
        
        Now we check whether various elements are in `G_1` and
        `G_2`::
        
            sage: G2.0 in G1
            True
            sage: G2.1 in G1
            False
            sage: G1.0 in G1
            True
            sage: G1.0 in G2
            True
        
        The integer `0` is in, since it coerces in::
        
            sage: 0 in G1
            True
        
        Elements that have a completely different ambient product Jacobian
        are never in `G`::
        
            sage: J0(23).cuspidal_subgroup().0 in G1
            False
            sage: J0(23).cuspidal_subgroup()(0) in G1
            False
        """
        try:
            self(x)
        except TypeError:
            return False
        return True

    def subgroup(self, gens):
        """
        Return the subgroup of self spanned by the given generators, which
        all must be elements of self.
        
        EXAMPLES::
        
            sage: J = J0(23)
            sage: G = J.torsion_subgroup(11); G
            Finite subgroup with invariants [11, 11, 11, 11] over QQ of Abelian variety J0(23) of dimension 2
        
        We create the subgroup of the 11-torsion subgroup of
        `J_0(23)` generated by the first `11`-torsion
        point::
        
            sage: H = G.subgroup([G.0]); H
            Finite subgroup with invariants [11] over QQbar of Abelian variety J0(23) of dimension 2
            sage: H.invariants()
            [11]
        
        We can also create a subgroup from a list of objects that coerce
        into the ambient rational homology.
        
        ::
        
            sage: H == G.subgroup([[1/11,0,0,0]])
            True
        """
        if not isinstance(gens, (tuple, list)):
            raise TypeError, "gens must be a list or tuple"
        A = self.abelian_variety()
        lattice = A._ambient_lattice().span([self(g).element() for g in gens])
        return FiniteSubgroup_lattice(self.abelian_variety(), lattice, field_of_definition=QQbar)

    def invariants(self):
        r"""
        Return elementary invariants of this abelian group, by which we
        mean a nondecreasing (immutable) sequence of integers
        `n_i`, `1 \leq i \leq k`, with `n_i`
        dividing `n_{i+1}`, and such that this group is abstractly
        isomorphic to
        `\ZZ/n_1\ZZ \times\cdots\times \ZZ/n_k\ZZ.`
        
        EXAMPLES::
        
            sage: J = J0(38)
            sage: C = J.cuspidal_subgroup(); C
            Finite subgroup with invariants [3, 45] over QQ of Abelian variety J0(38) of dimension 4
            sage: v = C.invariants(); v
            [3, 45]
            sage: v[0] = 5
            Traceback (most recent call last):
            ...
            ValueError: object is immutable; please change a copy instead.
            sage: type(v[0])
            <type 'sage.rings.integer.Integer'>
        
        ::
        
            sage: C * 3
            Finite subgroup with invariants [15] over QQ of Abelian variety J0(38) of dimension 4
        
        An example involving another cuspidal subgroup::
        
            sage: C = J0(22).cuspidal_subgroup(); C
            Finite subgroup with invariants [5, 5] over QQ of Abelian variety J0(22) of dimension 2
            sage: C.lattice()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [1/5 1/5 4/5   0]
            [  0   1   0   0]
            [  0   0   1   0]
            [  0   0   0 1/5]
            sage: C.invariants()
            [5, 5]
        """
        try:
            return self.__invariants
        except AttributeError:
            pass
        M = self.lattice().coordinate_module(self.abelian_variety().lattice())
        E = M.basis_matrix().change_ring(ZZ).elementary_divisors()
        v = [Integer(x) for x in E if x != 1]
        I = Sequence(v)
        I.sort()
        I.set_immutable()
        self.__invariants = I
        return I
        
class FiniteSubgroup_lattice(FiniteSubgroup):
    def __init__(self, abvar, lattice, field_of_definition=QQbar, check=True):
        """
        A finite subgroup of a modular abelian variety that is defined by a
        given lattice.
        
        INPUT:
        
        
        -  ``abvar`` - a modular abelian variety
        
        -  ``lattice`` - a lattice that contains the lattice of
           abvar
        
        -  ``field_of_definition`` - the field of definition
           of this finite group scheme
        
        -  ``check`` - bool (default: True) whether or not to
           check that lattice contains the abvar lattice.
        
        
        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
        """
        if check:
            if not is_FreeModule(lattice) or lattice.base_ring() != ZZ:
                raise TypeError, "lattice must be a free module over ZZ"
            if not abelian_variety.is_ModularAbelianVariety(abvar):
                raise TypeError, "abvar must be a modular abelian variety"
            if not abvar.lattice().is_submodule(lattice):
                lattice += abvar.lattice()
            if lattice.rank() != abvar.lattice().rank():
                raise ValueError, "lattice must contain the lattice of abvar with finite index"
        FiniteSubgroup.__init__(self, abvar, field_of_definition)
        self.__lattice = lattice

    def lattice(self):
        r"""
        Return lattice that defines this finite subgroup.
        
        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.lattice()
            Free module of degree 2 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1/3   0]
            [  0 1/5]
        """
        return self.__lattice
        
###########################################################################
# Elements of finite order
###########################################################################

class TorsionPoint(ModuleElement):
    def __init__(self, parent, element, check=True):
        """
        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 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.TorsionPoint'>
        """
        ModuleElement.__init__(self, parent)
        if check:
            if not element in parent.abelian_variety().vector_space():
                raise TypeError, "element must be a vector in the abelian variety's rational homology (embedded in the ambient Jacobian product)"
        if element.denominator() == 1:
            element = element.parent().zero_vector()
        self.__element = element

    def element(self):
        """
        Return an underlying QQ-vector space element that defines this
        element of a modular abelian variety. This is a vector in the
        ambient Jacobian variety's rational homology.
        
        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'>
        """
        return self.__element

    def _repr_(self):
        r"""
        Return string representation of this finite subgroup element. 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)]'
        """
        return '[%s]'%self.__element

    def _add_(self, other):
        """
        Add two finite subgroup elements with the same parent. This is
        called implicitly by +.
        
        INPUT:
        
        
        -  ``other`` - a TorsionPoint with the same parent as
           self
        
        
        OUTPUT: a TorsionPoint
        
        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)]
        """
        return TorsionPoint(self.parent(), self.__element + other.__element, check=False)
    
    def _sub_(self, other):
        """
        Subtract two finite subgroup elements with the same parent. This is
        called implicitly by +.
        
        INPUT:
        
        
        -  ``other`` - a TorsionPoint with the same parent as
           self
        
        
        OUTPUT: a TorsionPoint
        
        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)]
        """
        return TorsionPoint(self.parent(), self.__element - other.__element, check=False)

    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)]
        """
        return TorsionPoint(self.parent(), -self.__element, check=False)

    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)]
        """
        return TorsionPoint(self.parent(), ZZ(left) * self.__element, check=False)

    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)]
        """
        return TorsionPoint(self.parent(), self.__element * right, check=False)

    def __cmp__(self, right):
        """
        Compare self and right.
        
        INPUT:
        
        
        -  ``self, right`` - elements of the same finite
           abelian variety subgroup.
        
        
        OUTPUT: -1, 0, or 1
        
        EXAMPLES::
        
            sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]])
            sage: cmp(G.0, G.1)
            1
            sage: cmp(G.0, G.0)
            0
            sage: 3*G.0 == 0
            True
            sage: 3*G.0 == 5*G.1
            True
        
        We make sure things that shouldn't be equal aren't::
        
            sage: H = J0(14).finite_subgroup([[1/3,0]])
            sage: G.0 == H.0
            False            
            sage: cmp(G.0, H.0)
            -1
            sage: G.0
            [(1/3, 0)]
            sage: H.0
            [(1/3, 0)]
        """
        if not isinstance(right, TorsionPoint):
            return cmp(type(self), type(right))
        A = self.parent().abelian_variety()
        B = right.parent().abelian_variety()
        if A.groups() != B.groups():
            return cmp(A,B)
        elif self.__element.change_ring(QQ) - right.__element.change_ring(QQ) in A.lattice() + B.lattice():
            return 0
        else:
            return cmp(self.__element, right.__element)
        
    def additive_order(self):
        """
        Return the additive order of this element.
        
        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
        """
        return self._relative_element().denominator()

    def _relative_element(self):
        """
        Return coordinates of this element in terms of 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)
        """
        return self.parent().abelian_variety().lattice().coordinate_vector(self.__element)