def reduce_cusp(self, c):
        r"""
        Calculate the unique reduced representative of the equivalence of the
        cusp `c` modulo this group. The reduced representative of an
        equivalence class is the unique cusp in the class of the form `u/v`
        with `u, v \ge 0` coprime, `v` minimal, and `u` minimal for that `v`.

        EXAMPLES::

            sage: Gamma(5).reduce_cusp(1/5)
            Infinity
            sage: Gamma(5).reduce_cusp(7/8)
            3/2
            sage: Gamma(6).reduce_cusp(4/3)
            2/3

        TESTS::

            sage: G = Gamma(50); all([c == G.reduce_cusp(c) for c in G.cusps()])
            True
        """
        N = self.level()
        c = Cusp(c)
        u,v = c.numerator() % N, c.denominator() % N
        if (v > N//2) or (2*v == N and u > N//2): 
            u,v = -u,-v
        u,v = _lift_pair(u,v,N)
        return Cusp(u,v)

    def cusp_data(self, c):
        r"""
        Return a triple (g, w, t) where g is an element of self generating the
        stabiliser of the given cusp, w is the width of the cusp, and t is 1 if
        the cusp is regular and -1 if not.

        EXAMPLES::

            sage: Gamma1(4).cusp_data(Cusps(1/2))
            (
            [ 1 -1]
            [ 4 -3], 1, -1
            )
        """
        c = Cusp(c)

        # first find an element of SL2Z sending infinity to the given cusp
        w = lift_to_sl2z(c.denominator(), c.numerator(), 0)
        g = SL2Z([w[3], w[1], w[2],w[0]])

        for d in xrange(1,1+self.index()):
            if g * SL2Z([1,d,0,1]) * (~g) in self:
                return (g * SL2Z([1,d,0,1]) * (~g), d, 1)
            elif g * SL2Z([-1,-d,0,-1]) * (~g) in self:
                return (g * SL2Z([-1,-d,0,-1]) * (~g), d, -1)
        raise ArithmeticError("Can't get here!")

    def _find_cusps(self):
        r"""
        Calculate a list of inequivalent cusps.

        EXAMPLES::

            sage: sage.modular.arithgroup.congroup_generic.CongruenceSubgroup(5)._find_cusps()
            Traceback (most recent call last):
            ...
            NotImplementedError

        NOTE: There is a generic algorithm implemented at the top level that
        uses the coset representatives of self. This is *very slow* and for all
        the standard congruence subgroups there is a quicker way of doing it,
        so this should usually be overridden in subclasses; but it doesn't have
        to be.
        """
        i = Cusp([1,0])
        L = [i]
        for a in self.coset_reps():
            ai = i.apply([a.a(), a.b(), a.c(), a.d()])
            new = 1
            for v in L:
                if self.are_equivalent(ai, v):
                    new = 0
                    break
            if new == 1:
                L.append(ai)
        return L

    def are_equivalent(self, x, y, trans = False):
        r""" 
        Test whether or not cusps x and y are equivalent modulo self.  If self
        has a reduce_cusp() method, use that; otherwise do a slow explicit
        test. 

        If trans = False, returns True or False. If trans = True, then return
        either False or an element of self mapping x onto y.

        EXAMPLE::

            sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(0), trans=True)
            [  3  -1]
            [-14   5]
            sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(1/7))
            False
        """
        x = Cusp(x)
        y = Cusp(y)
        if not trans:
            try:
                xr = self.reduce_cusp(x)
                yr = self.reduce_cusp(y)
                if xr != yr:
                    return False
                if xr == yr:
                    return True
            except NotImplementedError:
                pass

        from all import SL2Z 
    
        vx = lift_to_sl2z(x.numerator(),x.denominator(), 0)
        dx = SL2Z([vx[2], -vx[0], vx[3], -vx[1]])
        vy = lift_to_sl2z(y.numerator(),y.denominator(), 0)
        dy = SL2Z([vy[2], -vy[0], vy[3], -vy[1]])
    
        for i in xrange(self.index()):
            # Note that the width of any cusp is bounded above by the index of self.
            # If self is congruence, then the level of self is a much better bound, but
            # this method is written to work with non-congruence subgroups as well,
            if dy * SL2Z([1,i,0,1])*(~dx) in self:
                if trans:
                    return dy * SL2Z([1,i,0,1]) * ~dx
                else:
                    return True
            elif (self.is_odd() and dy * SL2Z([-1,-i,0,-1]) * ~dx in self):
                if trans:
                    return dy * SL2Z([-1,-i,0,-1]) * ~dx
                else:
                    return True
        return False

    def _compute_lattice(self, rational_only=False, rational_subgroup=False):
        r"""
        Return a list of vectors that define elements of the rational
        homology that generate this finite subgroup.
        
        INPUT:
        
        
        -  ``rational_only`` - bool (default: False); if
           ``True``, only use rational cusps.
        
        
        OUTPUT:
        
        
        -  ``list`` - list of vectors
        
        
        EXAMPLES::
        
            sage: J = J0(37)
            sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
            sage: C._compute_lattice()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [  1   0   0   0]
            [  0   1   0   0]
            [  0   0   1   0]
            [  0   0   0 1/3]
            sage: J = J0(43)
            sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
            sage: C._compute_lattice()
            Free module of degree 6 and rank 6 over Integer Ring
            Echelon basis matrix:
            [  1   0   0   0   0   0]
            [  0 1/7   0 6/7   0 5/7]
            [  0   0   1   0   0   0]
            [  0   0   0   1   0   0]
            [  0   0   0   0   1   0]
            [  0   0   0   0   0   1]
            sage: J = J0(22)
            sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
            sage: C._compute_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: J = J1(13)
            sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
            sage: C._compute_lattice()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [ 1/19     0     0  9/19]
            [    0  1/19  1/19 18/19]
            [    0     0     1     0]
            [    0     0     0     1]
        
        We compute with and without the optional
        ``rational_only`` option.
        
        ::
        
            sage: J = J0(27); G = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
            sage: G._compute_lattice()
            Free module of degree 2 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1/3   0]
            [  0 1/3]
            sage: G._compute_lattice(rational_only=True)
            Free module of degree 2 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1/3   0]
            [  0   1]
        """
        A = self.abelian_variety()
        Cusp = A.modular_symbols()
        Amb  = Cusp.ambient_module()
        Eis  = Amb.eisenstein_submodule()

        C = Amb.cusps()
        N = Amb.level()
        
        if rational_subgroup:
            # QQ-rational subgroup of cuspidal subgroup
            assert A.is_ambient()
            Q = Cusp.abvarquo_rational_cuspidal_subgroup()
            return Q.V()
        
        if rational_only:
            # subgroup generated by differences of rational cusps
            if not is_Gamma0(A.group()):
                raise NotImplementedError, 'computation of rational cusps only implemented in Gamma0 case.'
            if not N.is_squarefree():
                data = [n for n in range(2,N) if gcd(n,N) == 1]
                C = [c for c in C if is_rational_cusp_gamma0(c, N, data)]

        v = [Amb([infinity, alpha]).element() for alpha in C]
        cusp_matrix = matrix(QQ, len(v), Amb.dimension(), v)
#.........这里部分代码省略.........

    def _reduce_cusp(self, c):
        r"""
        Compute a minimal representative for the given cusp c.

        Returns a pair (c', t), where c' is the minimal representative
        for the given cusp, and t is either 1 or -1, as explained
        below. Largely for internal use.

        The minimal representative for a cusp is the element in `P^1(Q)`
        in lowest terms with minimal positive denominator, and minimal
        positive numerator for that denominator.

        Two cusps `u1/v1` and `u2/v2` are equivalent modulo `\Gamma_H(N)`
        if and only if

        - `v1 =  h*v2 (mod N)` and `u1 =  h^(-1)*u2 (mod gcd(v1,N))`

        or

        - `v1 = -h*v2 (mod N)` and `u1 = -h^(-1)*u2 (mod gcd(v1,N))`

        for some `h \in H`. Then t is 1 or -1 as c and c' fall into
        the first or second case, respectively.

        EXAMPLES::

            sage: GammaH(6,[5])._reduce_cusp(Cusp(5,3))
            (1/3, -1)
            sage: GammaH(12,[5])._reduce_cusp(Cusp(8,9))
            (1/3, -1)
            sage: GammaH(12,[5])._reduce_cusp(Cusp(5,12))
            (Infinity, 1)
            sage: GammaH(12,[])._reduce_cusp(Cusp(5,12))
            (5/12, 1)
            sage: GammaH(21,[5])._reduce_cusp(Cusp(-9/14))
            (1/7, 1)
        """
        c = Cusp(c)
        N = int(self.level())
        Cusps = c.parent()
        v = int(c.denominator() % N)
        H = self._list_of_elements_in_H()

        # First, if N | v, take care of this case. If u is in \pm H,
        # then we return Infinity. If not, let u_0 be the minimum
        # of \{ h*u | h \in \pm H \}. Then return u_0/N.
        if not v:
            u = c.numerator() % N
            if u in H:
                return Cusps((1,0)), 1
            if (N-u) in H:
                return Cusps((1,0)), -1
            ls = [ (u*h)%N for h in H ]
            m1 = min(ls)
            m2 = N-max(ls)
            if m1 < m2:
                return Cusps((m1,N)), 1
            else:
                return Cusps((m2,N)), -1

        u = int(c.numerator() % v)
        gcd = get_gcd(N)
        d = gcd(v,N)

        # If (N,v) == 1, let v_0 be the minimal element
        # in \{ v * h | h \in \pm H \}. Then we either return
        # Infinity or 1/v_0, as v is or is not in \pm H,
        # respectively.
        if d == 1:
            if v in H:
                return Cusps((0,1)), 1
            if (N-v) in H:
                return Cusps((0,1)), -1
            ls = [ (v*h)%N for h in H ]
            m1 = min(ls)
            m2 = N-max(ls)
            if m1 < m2:
                return Cusps((1,m1)), 1
            else:
                return Cusps((1,m2)), -1

        val_min = v
        inv_mod = get_inverse_mod(N)

        # Now we're in the case (N,v) > 1. So we have to do several
        # steps: first, compute v_0 as above. While computing this
        # minimum, keep track of *all* pairs of (h,s) which give this
        # value of v_0.
        hs_ls = [(1,1)]
        for h in H:
            tmp = (v*h)%N

            if tmp < val_min:
                val_min = tmp
                hs_ls = [(inv_mod(h,N), 1)]
            elif tmp == val_min:
                hs_ls.append((inv_mod(h,N), 1))

            if (N-tmp) < val_min:
                val_min = N - tmp
#.........这里部分代码省略.........