quaternion_algebra.py

"""
Quaternion Algebras

AUTHORS:

- Jon Bobber (2009): rewrite

- William Stein (2009): rewrite

- Julian Rueth (2014-03-02): use UniqueFactory for caching

This code is partly based on Sage code by David Kohel from 2005.

TESTS:

Pickling test::

    sage: Q.<i,j,k> = QuaternionAlgebra(QQ,-5,-2)
    sage: Q == loads(dumps(Q))
    True
"""

# ****************************************************************************
#       Copyright (C) 2009 William Stein <wstein@gmail.com>
#       Copyright (C) 2009 Jonathan Bober <jwbober@gmail.com>
#       Copyright (C) 2014 Julian Rueth <julian.rueth@fsfe.org>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************
from __future__ import print_function, absolute_import

from sage.arith.all import (hilbert_conductor_inverse, hilbert_conductor,
        factor, gcd, kronecker_symbol, valuation)
from sage.rings.all import RR, Integer
from sage.rings.integer_ring import ZZ
from sage.rings.rational import Rational
from sage.rings.finite_rings.finite_field_constructor import GF

from sage.rings.ring import Algebra
from sage.rings.ideal import Ideal_fractional
from sage.rings.rational_field import is_RationalField, QQ
from sage.rings.infinity import infinity
from sage.rings.number_field.number_field import is_NumberField
from sage.rings.power_series_ring import PowerSeriesRing
from sage.structure.category_object import normalize_names
from sage.structure.parent_gens import ParentWithGens
from sage.structure.parent import Parent
from sage.matrix.matrix_space import MatrixSpace
from sage.matrix.constructor import diagonal_matrix, matrix
from sage.structure.sequence import Sequence
from sage.structure.element import is_RingElement
from sage.structure.factory import UniqueFactory
from sage.modules.free_module import VectorSpace, FreeModule
from sage.modules.free_module_element import vector

from operator import itemgetter

from . import quaternion_algebra_element
from . import quaternion_algebra_cython

from sage.modular.modsym.p1list import P1List

from sage.misc.cachefunc import cached_method

from sage.categories.fields import Fields
from sage.categories.algebras import Algebras
_Fields = Fields()

########################################################
# Constructor
########################################################


class QuaternionAlgebraFactory(UniqueFactory):
    r"""
    There are three input formats:

    - ``QuaternionAlgebra(a, b)``: quaternion algebra generated by ``i``, ``j``
      subject to `i^2 = a`, `j^2 = b`, `j \cdot i = -i \cdot j`.

    - ``QuaternionAlgebra(K, a, b)``: same as above but over a field ``K``.
      Here, ``a`` and ``b`` are nonzero elements of a field (``K``) of
      characteristic not 2, and we set `k = i \cdot j`.

    - ``QuaternionAlgebra(D)``: a rational quaternion algebra with
      discriminant ``D``, where `D > 1` is a squarefree integer.

    EXAMPLES:

    ``QuaternionAlgebra(a, b)`` - return quaternion algebra over the
    *smallest* field containing the nonzero elements ``a`` and ``b`` with
    generators ``i``, ``j``, ``k`` with `i^2=a`, `j^2=b` and `j \cdot i =
    -i \cdot j`::

        sage: QuaternionAlgebra(-2,-3)
        Quaternion Algebra (-2, -3) with base ring Rational Field
        sage: QuaternionAlgebra(GF(5)(2), GF(5)(3))
        Quaternion Algebra (2, 3) with base ring Finite Field of size 5
        sage: QuaternionAlgebra(2, GF(5)(3))
        Quaternion Algebra (2, 3) with base ring Finite Field of size 5
        sage: QuaternionAlgebra(QQ[sqrt(2)](-1), -5)
        Quaternion Algebra (-1, -5) with base ring Number Field in sqrt2 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?
        sage: QuaternionAlgebra(sqrt(-1), sqrt(-3))
        Quaternion Algebra (I, sqrt(-3)) with base ring Symbolic Ring
        sage: QuaternionAlgebra(1r,1)
        Quaternion Algebra (1, 1) with base ring Rational Field

    Python ints, longs and floats may be passed to the
    ``QuaternionAlgebra(a, b)`` constructor, as may  all pairs of nonzero
    elements of a ring not of characteristic 2. The following tests address
    the issues raised in :trac:`10601`::

        sage: QuaternionAlgebra(1r,1)
        Quaternion Algebra (1, 1) with base ring Rational Field
        sage: QuaternionAlgebra(1,1.0r)
        Quaternion Algebra (1.00000000000000, 1.00000000000000) with base ring Real Field with 53 bits of precision
        sage: QuaternionAlgebra(0,0)
        Traceback (most recent call last):
        ...
        ValueError: a and b must be nonzero
        sage: QuaternionAlgebra(GF(2)(1),1)
        Traceback (most recent call last):
        ...
        ValueError: a and b must be elements of a ring with characteristic not 2
        sage: a = PermutationGroupElement([1,2,3])
        sage: QuaternionAlgebra(a, a)
        Traceback (most recent call last):
        ...
        ValueError: a and b must be elements of a ring with characteristic not 2

    ``QuaternionAlgebra(K, a, b)`` - return quaternion algebra over the
    field ``K`` with generators ``i``, ``j``, ``k`` with `i^2=a`, `j^2=b`
    and `i \cdot j = -j \cdot i`::

        sage: QuaternionAlgebra(QQ, -7, -21)
        Quaternion Algebra (-7, -21) with base ring Rational Field
        sage: QuaternionAlgebra(QQ[sqrt(2)], -2,-3)
        Quaternion Algebra (-2, -3) with base ring Number Field in sqrt2 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?

    ``QuaternionAlgebra(D)`` - ``D`` is a squarefree integer; returns a
    rational quaternion algebra of discriminant ``D``::

        sage: QuaternionAlgebra(1)
        Quaternion Algebra (-1, 1) with base ring Rational Field
        sage: QuaternionAlgebra(2)
        Quaternion Algebra (-1, -1) with base ring Rational Field
        sage: QuaternionAlgebra(7)
        Quaternion Algebra (-1, -7) with base ring Rational Field
        sage: QuaternionAlgebra(2*3*5*7)
        Quaternion Algebra (-22, 210) with base ring Rational Field

    If the coefficients `a` and `b` in the definition of the quaternion
    algebra are not integral, then a slower generic type is used for
    arithmetic::

        sage: type(QuaternionAlgebra(-1,-3).0)
        <... 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field'>
        sage: type(QuaternionAlgebra(-1,-3/2).0)
        <... 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic'>

    Make sure caching is sane::

        sage: A = QuaternionAlgebra(2,3); A
        Quaternion Algebra (2, 3) with base ring Rational Field
        sage: B = QuaternionAlgebra(GF(5)(2),GF(5)(3)); B
        Quaternion Algebra (2, 3) with base ring Finite Field of size 5
        sage: A is QuaternionAlgebra(2,3)
        True
        sage: B is QuaternionAlgebra(GF(5)(2),GF(5)(3))
        True
        sage: Q = QuaternionAlgebra(2); Q
        Quaternion Algebra (-1, -1) with base ring Rational Field
        sage: Q is QuaternionAlgebra(QQ,-1,-1)
        True
        sage: Q is QuaternionAlgebra(-1,-1)
        True
        sage: Q.<ii,jj,kk> = QuaternionAlgebra(15); Q.variable_names()
        ('ii', 'jj', 'kk')
        sage: QuaternionAlgebra(15).variable_names()
        ('i', 'j', 'k')

    TESTS:

    Verify that bug found when working on :trac:`12006` involving coercing
    invariants into the base field is fixed::

        sage: Q = QuaternionAlgebra(-1,-1); Q
        Quaternion Algebra (-1, -1) with base ring Rational Field
        sage: parent(Q._a)
        Rational Field
        sage: parent(Q._b)
        Rational Field
    """
    def create_key(self, arg0, arg1=None, arg2=None, names='i,j,k'):
        """
        Create a key that uniquely determines a quaternion algebra.

        TESTS::

            sage: QuaternionAlgebra.create_key(-1,-1)
            (Rational Field, -1, -1, ('i', 'j', 'k'))
        """
        # QuaternionAlgebra(D)
        if arg1 is None and arg2 is None:
            K = QQ
            D = Integer(arg0)
            a, b = hilbert_conductor_inverse(D)
            a = Rational(a)
            b = Rational(b)

        elif arg2 is None:
            # If arg0 or arg1 are Python data types, coerce them
            # to the relevant Sage types. This is a bit inelegant.
            L = []
            for a in [arg0,arg1]:
                if is_RingElement(a):
                    L.append(a)
                elif isinstance(a, int):
                    L.append(Integer(a))
                elif isinstance(a, float):
                    L.append(RR(a))
                else:
                    raise ValueError("a and b must be elements of a ring with characteristic not 2")

            # QuaternionAlgebra(a, b)
            v = Sequence(L)
            K = v.universe().fraction_field()
            a = K(v[0])
            b = K(v[1])

        # QuaternionAlgebra(K, a, b)
        else:
            K = arg0
            if K not in _Fields:
                raise TypeError("base ring of quaternion algebra must be a field")
            a = K(arg1)
            b = K(arg2)

        if K.characteristic() == 2:
            # Lameness!
            raise ValueError("a and b must be elements of a ring with characteristic not 2")
        if a == 0 or b == 0:
            raise ValueError("a and b must be nonzero")

        names = normalize_names(3, names)
        return (K, a, b, names)

    def create_object(self, version, key, **extra_args):
        """
        Create the object from the key (extra arguments are ignored). This is
        only called if the object was not found in the cache.

        TESTS::

            sage: QuaternionAlgebra.create_object("6.0", (QQ, -1, -1, ('i', 'j', 'k')))
            Quaternion Algebra (-1, -1) with base ring Rational Field

        """
        K, a, b, names = key
        return QuaternionAlgebra_ab(K, a, b, names=names)

QuaternionAlgebra = QuaternionAlgebraFactory("QuaternionAlgebra")

########################################################
# Classes
########################################################

def is_QuaternionAlgebra(A):
    """
    Return ``True`` if ``A`` is of the QuaternionAlgebra data type.

    EXAMPLES::

        sage: sage.algebras.quatalg.quaternion_algebra.is_QuaternionAlgebra(QuaternionAlgebra(QQ,-1,-1))
        True
        sage: sage.algebras.quatalg.quaternion_algebra.is_QuaternionAlgebra(ZZ)
        False
    """
    return isinstance(A, QuaternionAlgebra_abstract)


class QuaternionAlgebra_abstract(Algebra):
    def _repr_(self):
        """
        EXAMPLES::

            sage: sage.algebras.quatalg.quaternion_algebra.QuaternionAlgebra_abstract(QQ)._repr_()
            'Quaternion Algebra with base ring Rational Field'
        """
        return "Quaternion Algebra with base ring %s" % self.base_ring()

    def ngens(self):
        """
        Return the number of generators of the quaternion algebra as a K-vector
        space, not including 1.

        This value is always 3: the algebra is spanned
        by the standard basis `1`, `i`, `j`, `k`.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ,-5,-2)
            sage: Q.ngens()
            3
            sage: Q.gens()
            [i, j, k]
        """
        return 3

    def basis(self):
        """
        Return the fixed basis of ``self``, which is `1`, `i`, `j`, `k`, where
        `i`, `j`, `k` are the generators of ``self``.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ,-5,-2)
            sage: Q.basis()
            (1, i, j, k)

            sage: Q.<xyz,abc,theta> = QuaternionAlgebra(GF(9,'a'),-5,-2)
            sage: Q.basis()
            (1, xyz, abc, theta)

        The basis is cached::

            sage: Q.basis() is Q.basis()
            True
        """
        try:
            return self.__basis
        except AttributeError:
            self.__basis = tuple([self(1)] + list(self.gens()))
            return self.__basis

    @cached_method
    def inner_product_matrix(self):
        """
        Return the inner product matrix associated to ``self``.

        This is the
        Gram matrix of the reduced norm as a quadratic form on ``self``.
        The standard basis `1`, `i`, `j`, `k` is orthogonal, so this matrix
        is just the diagonal matrix with diagonal entries `2`, `2a`, `2b`,
        `2ab`.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(-5,-19)
            sage: Q.inner_product_matrix()
            [  2   0   0   0]
            [  0  10   0   0]
            [  0   0  38   0]
            [  0   0   0 190]
        """
        a, b = self._a, self._b
        M = diagonal_matrix(self.base_ring(), [2, -2 * a, -2 * b, 2 * a * b])
        M.set_immutable()
        return M

    def is_commutative(self):
        """
        Return ``False`` always, since all quaternion algebras are
        noncommutative.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3,-7)
            sage: Q.is_commutative()
            False
        """
        return False

    def is_division_algebra(self):
        """
        Return ``True`` if the quaternion algebra is a division algebra (i.e.
        every nonzero element in ``self`` is invertible), and ``False`` if the
        quaternion algebra is isomorphic to the 2x2 matrix algebra.

        EXAMPLES::

            sage: QuaternionAlgebra(QQ,-5,-2).is_division_algebra()
            True
            sage: QuaternionAlgebra(1).is_division_algebra()
            False
            sage: QuaternionAlgebra(2,9).is_division_algebra()
            False
            sage: QuaternionAlgebra(RR(2.),1).is_division_algebra()
            Traceback (most recent call last):
            ...
            NotImplementedError: base field must be rational numbers
        """
        if not is_RationalField(self.base_ring()):
            raise NotImplementedError("base field must be rational numbers")
        return self.discriminant() != 1

    def is_matrix_ring(self):
        """
        Return ``True`` if the quaternion algebra is isomorphic to the 2x2
        matrix ring, and ``False`` if ``self`` is a division algebra (i.e.
        every nonzero element in ``self`` is invertible).

        EXAMPLES::

            sage: QuaternionAlgebra(QQ,-5,-2).is_matrix_ring()
            False
            sage: QuaternionAlgebra(1).is_matrix_ring()
            True
            sage: QuaternionAlgebra(2,9).is_matrix_ring()
            True
            sage: QuaternionAlgebra(RR(2.),1).is_matrix_ring()
            Traceback (most recent call last):
            ...
            NotImplementedError: base field must be rational numbers

        """
        if not is_RationalField(self.base_ring()):
            raise NotImplementedError("base field must be rational numbers")
        return self.discriminant() == 1

    def is_exact(self):
        """
        Return ``True`` if elements of this quaternion algebra are represented
        exactly, i.e. there is no precision loss when doing arithmetic. A
        quaternion algebra is exact if and only if its base field is
        exact.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3, -7)
            sage: Q.is_exact()
            True
            sage: Q.<i,j,k> = QuaternionAlgebra(Qp(7), -3, -7)
            sage: Q.is_exact()
            False
        """
        return self.base_ring().is_exact()

    def is_field(self, proof = True):
        """
        Return ``False`` always, since all quaternion algebras are
        noncommutative and all fields are commutative.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3, -7)
            sage: Q.is_field()
            False
        """
        return False

    def is_finite(self):
        """
        Return ``True`` if the quaternion algebra is finite as a set.

        Algorithm: A quaternion algebra is finite if and only if the
        base field is finite.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3, -7)
            sage: Q.is_finite()
            False
            sage: Q.<i,j,k> = QuaternionAlgebra(GF(5), -3, -7)
            sage: Q.is_finite()
            True
        """
        return self.base_ring().is_finite()

    def is_integral_domain(self, proof=True):
        """
        Return ``False`` always, since all quaternion algebras are
        noncommutative and integral domains are commutative (in Sage).

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3, -7)
            sage: Q.is_integral_domain()
            False
        """
        return False

    def is_noetherian(self):
        """
        Return ``True`` always, since any quaternion algebra is a noetherian
        ring (because it is a finitely generated module over a field).

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3, -7)
            sage: Q.is_noetherian()
            True
        """
        return True

    def order(self):
        """
        Return the number of elements of the quaternion algebra, or
        ``+Infinity`` if the algebra is not finite.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -3, -7)
            sage: Q.order()
            +Infinity
            sage: Q.<i,j,k> = QuaternionAlgebra(GF(5), -3, -7)
            sage: Q.order()
            625
        """
        return (self.base_ring().order())**4

    def random_element(self, *args, **kwds):
        """
        Return a random element of this quaternion algebra.

        The ``args`` and ``kwds`` are passed to the ``random_element`` method
        of the base ring.

        EXAMPLES::

            sage: g = QuaternionAlgebra(QQ[sqrt(2)], -3, 7).random_element()
            sage: g.parent() is QuaternionAlgebra(QQ[sqrt(2)], -3, 7)
            True
            sage: g = QuaternionAlgebra(-3, 19).random_element()
            sage: g.parent() is QuaternionAlgebra(-3, 19)
            True
            sage: g = QuaternionAlgebra(GF(17)(2), 3).random_element()
            sage: g.parent() is QuaternionAlgebra(GF(17)(2), 3)
            True

        Specify the numerator and denominator bounds::

            sage: g = QuaternionAlgebra(-3,19).random_element(10^6, 10^6)
            sage: for h in g:
            ....:     assert h.numerator() in range(-10^6, 10^6 + 1)
            ....:     assert h.denominator() in range(10^6 + 1)

            sage: g = QuaternionAlgebra(-3,19).random_element(5, 4)
            sage: for h in g:
            ....:     assert h.numerator() in range(-5, 5 + 1)
            ....:     assert h.denominator() in range(4 + 1)
        """
        K = self.base_ring()
        return self([K.random_element(*args, **kwds) for _ in range(4)])

    @cached_method
    def vector_space(self):
        """
        Return the vector space associated to ``self`` with inner product given
        by the reduced norm.

        EXAMPLES::

            sage: QuaternionAlgebra(-3,19).vector_space()
            Ambient quadratic space of dimension 4 over Rational Field
            Inner product matrix:
            [   2    0    0    0]
            [   0    6    0    0]
            [   0    0  -38    0]
            [   0    0    0 -114]
        """
        return VectorSpace(self.base_ring(), 4, inner_product_matrix=self.inner_product_matrix())


class QuaternionAlgebra_ab(QuaternionAlgebra_abstract):
    """
    The quaternion algebra of the form `(a, b/K)`, where `i^2=a`, `j^2 = b`,
    and `j*i = -i*j`.  ``K`` is a field not of characteristic 2 and ``a``,
    ``b`` are nonzero elements of ``K``.

    See ``QuaternionAlgebra`` for many more examples.

    INPUT:

    - ``base_ring`` -- commutative ring
    - ``a, b`` -- elements of ``base_ring``
    - ``names`` -- string (optional, default 'i,j,k') names of the generators

    EXAMPLES::

        sage: QuaternionAlgebra(QQ, -7, -21)  # indirect doctest
        Quaternion Algebra (-7, -21) with base ring Rational Field
    """
    def __init__(self, base_ring, a, b, names='i,j,k'):
        """
        Create the quaternion algebra with `i^2 = a`, `j^2 = b`, and
        `i*j = -j*i = k`.

        TESTS:

        Test making quaternion elements (using the element constructor)::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ,-1,-2)
            sage: a = Q(2/3); a
            2/3
            sage: type(a)
            <... 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field'>
            sage: Q(a)
            2/3
            sage: Q([1,2,3,4])
            1 + 2*i + 3*j + 4*k
            sage: Q((1,2,3,4))
            1 + 2*i + 3*j + 4*k
            sage: Q(-3/5)
            -3/5

            sage: TestSuite(Q).run()

        The base ring must be a field::

            sage: Q.<ii,jj,kk> = QuaternionAlgebra(ZZ,-5,-19)
            Traceback (most recent call last):
            ...
            TypeError: base ring of quaternion algebra must be a field
        """
        ParentWithGens.__init__(self, base_ring, names=names, category=Algebras(base_ring).Division())
        self._a = a
        self._b = b
        if is_RationalField(base_ring) and a.denominator() == 1 and b.denominator() == 1:
            self.Element = quaternion_algebra_element.QuaternionAlgebraElement_rational_field
        elif is_NumberField(base_ring) and base_ring.degree() > 2 and base_ring.is_absolute() and \
                 a.denominator() == 1 and b.denominator() == 1 and base_ring.defining_polynomial().is_monic():
            # This QuaternionAlgebraElement_number_field class is not
            # designed to work with elements of a quadratic field.  To
            # do that, the main thing would be to implement
            # __getitem__, etc.  This would maybe give a factor of 2
            # (or more?) speedup.  Much care must be taken because the
            # underlying representation of quadratic fields is a bit
            # tricky.
            self.Element = quaternion_algebra_element.QuaternionAlgebraElement_number_field
        elif base_ring in _Fields:
            self.Element = quaternion_algebra_element.QuaternionAlgebraElement_generic
        else:
            raise TypeError("base ring of quaternion algebra must be a field")
        self._populate_coercion_lists_(coerce_list=[base_ring])
        self._gens = [self([0,1,0,0]), self([0,0,1,0]), self([0,0,0,1])]

    @cached_method
    def maximal_order(self, take_shortcuts=True):
        r"""
        Return a maximal order in this quaternion algebra.

        The algorithm used is from [Voi2012]_.

        INPUT:

        - ``take_shortcuts`` -- (default: ``True``) if the discriminant is
          prime and the invariants of the algebra are of a nice form, use
          Proposition 5.2 of [Piz1980]_.

        OUTPUT:

        A maximal order in this quaternion algebra.

        EXAMPLES::

            sage: QuaternionAlgebra(-1,-7).maximal_order()
            Order of Quaternion Algebra (-1, -7) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)

            sage: QuaternionAlgebra(-1,-1).maximal_order().basis()
            (1/2 + 1/2*i + 1/2*j + 1/2*k, i, j, k)

            sage: QuaternionAlgebra(-1,-11).maximal_order().basis()
            (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)

            sage: QuaternionAlgebra(-1,-3).maximal_order().basis()
            (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)

            sage: QuaternionAlgebra(-3,-1).maximal_order().basis()
            (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)

            sage: QuaternionAlgebra(-2,-5).maximal_order().basis()
            (1/2 + 1/2*j + 1/2*k, 1/4*i + 1/2*j + 1/4*k, j, k)

            sage: QuaternionAlgebra(-5,-2).maximal_order().basis()
            (1/2 + 1/2*i - 1/2*k, 1/2*i + 1/4*j - 1/4*k, i, -k)

            sage: QuaternionAlgebra(-17,-3).maximal_order().basis()
            (1/2 + 1/2*j, 1/2*i + 1/2*k, -1/3*j - 1/3*k, k)

            sage: QuaternionAlgebra(-3,-17).maximal_order().basis()
            (1/2 + 1/2*i, 1/2*j - 1/2*k, -1/3*i + 1/3*k, -k)

            sage: QuaternionAlgebra(-17*9,-3).maximal_order().basis()
            (1, 1/3*i, 1/6*i + 1/2*j, 1/2 + 1/3*j + 1/18*k)

            sage: QuaternionAlgebra(-2, -389).maximal_order().basis()
            (1/2 + 1/2*j + 1/2*k, 1/4*i + 1/2*j + 1/4*k, j, k)

        If you want bases containing 1, switch off ``take_shortcuts``::

            sage: QuaternionAlgebra(-3,-89).maximal_order(take_shortcuts=False)
            Order of Quaternion Algebra (-3, -89) with base ring Rational Field with basis (1, 1/2 + 1/2*i, j, 1/2 + 1/6*i + 1/2*j + 1/6*k)

            sage: QuaternionAlgebra(1,1).maximal_order(take_shortcuts=False)    # Matrix ring
            Order of Quaternion Algebra (1, 1) with base ring Rational Field with basis (1, 1/2 + 1/2*i, j, 1/2*j + 1/2*k)

            sage: QuaternionAlgebra(-22,210).maximal_order(take_shortcuts=False)
            Order of Quaternion Algebra (-22, 210) with base ring Rational Field with basis (1, i, 1/2*i + 1/2*j, 1/2 + 17/22*i + 1/44*k)

            sage: for d in ( m for m in range(1, 750) if is_squarefree(m) ):        # long time (3s)
            ....:     A = QuaternionAlgebra(d)
            ....:     R = A.maximal_order(take_shortcuts=False)
            ....:     assert A.discriminant() == R.discriminant()

        We do not support number fields other than the rationals yet::

            sage: K = QuadraticField(5)
            sage: QuaternionAlgebra(K,-1,-1).maximal_order()
            Traceback (most recent call last):
            ...
            NotImplementedError: maximal order only implemented for rational quaternion algebras
        """
        if self.base_ring() != QQ:
            raise NotImplementedError("maximal order only implemented for rational quaternion algebras")

        d_A = self.discriminant()

        # The following only works over QQ if the discriminant is prime
        # and if the invariants are of the special form
        # (every quaternion algebra of prime discriminant has a representation
        #  of such a form though)
        a, b = self.invariants()
        if take_shortcuts and d_A.is_prime() and a in ZZ and b in ZZ:
            a = ZZ(a)
            b = ZZ(b)
            i,j,k = self.gens()

            # if necessary, try to swap invariants to match Pizer's paper
            if (a != -1 and b == -1) or (b == -2) \
               or (a != -1 and a != -2 and (-a) % 8 != 1):
                a, b = b, a
                i, j = j, i
                k = i*j

            basis = []
            if (a,b) == (-1,-1):
                basis = [(1+i+j+k)/2, i, j, k]
            elif a == -1 and (-b).is_prime() and ((-b) % 4 == 3):
                basis = [(1+j)/2, (i+k)/2, j, k]
            elif a == -2 and (-b).is_prime() and ((-b) % 8 == 5):
                basis = [(1+j+k)/2, (i+2*j+k)/4, j, k]
            elif (-a).is_prime() and (-b).is_prime():
                q = -b
                p = -a

                if q % 4 == 3 and kronecker_symbol(p,q) == -1:
                    a = 0
                    while (a*a*p + 1)%q != 0:
                        a += 1
                    basis = [(1+j)/2, (i+k)/2, -(j+a*k)/q, k]

            if basis:
                return self.quaternion_order(basis)

        # The following code should always work (over QQ)
        # Start with <1,i,j,k>
        R = self.quaternion_order([1] + self.gens())
        d_R = R.discriminant()

        e_new_gens = []

        # For each prime at which R is not yet maximal, make it bigger
        for (p,p_val) in d_R.factor():
            e = R.basis()
            while self.quaternion_order(e).discriminant().valuation(p) > d_A.valuation(p):
                # Compute a normalized basis at p
                f = normalize_basis_at_p(list(e), p)

                # Ensure the basis lies in R by clearing denominators
                # (this may make the order smaller at q != p)
                # Also saturate the basis (divide out p as far as possible)
                V = self.base_ring()**4
                A = matrix(self.base_ring(), 4, 4, [list(g) for g in e])

                e_n = []
                x_rows = A.solve_left(matrix([ V(vec.coefficient_tuple()) for (vec,val) in f ]), check=False).rows()
                denoms = [ x.denominator() for x in x_rows ]
                for i in range(4):
                    vec = f[i][0]
                    val = f[i][1]

                    v = (val/2).floor()
                    e_n.append(denoms[i] / p**(v) * vec)

                # for e_n to become p-saturated we still need to sort by
                # ascending valuation of the quadratic form
                lst = sorted(zip(e_n, [f[m][1].mod(2) for m in range(4)]),
                             key=itemgetter(1))
                e_n = list(next(zip(*lst)))

                # Final step: Enlarge the basis at p
                if p != 2:
                    # ensure that v_p(e_n[1]**2) = 0 by swapping basis elements
                    if ZZ(e_n[1]**2).valuation(p) != 0:
                        if ZZ(e_n[2]**2).valuation(p) == 0:
                            e_n[1], e_n[2] = e_n[2], e_n[1]
                        else:
                            e_n[1], e_n[3] = e_n[3], e_n[1]

                    a = ZZ(e_n[1]**2)
                    b = ZZ(e_n[2]**2)

                    if b.valuation(p) > 0:      # if v_p(b) = 0, then already p-maximal
                        F = ZZ.quo(p)
                        if F(a).is_square():
                            x = F(a).sqrt().lift()
                            if (x**2 - a).mod(p**2) == 0:  # make sure v_p(x**2 - a) = 1
                                x = x + p
                            g = 1/p*(x - e_n[1])*e_n[2]
                            e_n[2] = g
                            e_n[3] = e_n[1]*g

                else:   # p == 2
                    t = e_n[1].reduced_trace()
                    a = -e_n[1].reduced_norm()
                    b = ZZ(e_n[2]**2)

                    if t.valuation(p) == 0:
                        if b.valuation(p) > 0:
                            x = a
                            if (x**2 - t*x + a).mod(p**2) == 0: # make sure v_p(...) = 1
                                x = x + p
                            g = 1/p*(x - e_n[1])*e_n[2]
                            e_n[2] = g
                            e_n[3] = e_n[1]*g

                    else:   # t.valuation(p) > 0
                        (y,z,w) = maxord_solve_aux_eq(a, b, p)
                        g = 1/p*(1 + y*e_n[1] + z*e_n[2] + w*e_n[1]*e_n[2])
                        h = (z*b)*e_n[1] - (y*a)*e_n[2]
                        e_n[1:4] = [g,h,g*h]
                        if (1 - a*y**2 - b*z**2 + a*b*w**2).valuation(2) > 2:
                            e_n = basis_for_quaternion_lattice(list(e) + e_n[1:], reverse=True)

                # e_n now contains elements that locally at p give a bigger order,
                # but the basis may be messed up at other primes (it might not even
                # be an order). We will join them all together at the end
                e = e_n

            e_new_gens.extend(e[1:])

        e_new = basis_for_quaternion_lattice(list(R.basis()) + e_new_gens, reverse=True)
        return self.quaternion_order(e_new)

    def invariants(self):
        """
        Return the structural invariants `a`, `b` of this quaternion
        algebra: ``self`` is generated by `i`, `j` subject to
        `i^2 = a`, `j^2 = b` and `j*i = -i*j`.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(15)
            sage: Q.invariants()
            (-3, 5)
            sage: i^2
            -3
            sage: j^2
            5
        """
        return self._a, self._b

    def __eq__(self, other):
        """
        Compare self and other.

        EXAMPLES::

            sage: QuaternionAlgebra(-1,-7) == QuaternionAlgebra(-1,-7)
            True
            sage: QuaternionAlgebra(-1,-7) == QuaternionAlgebra(-1,-5)
            False
        """
        if not isinstance(other, QuaternionAlgebra_abstract):
            return False
        return (self.base_ring() == other.base_ring() and
                (self._a, self._b) == (other._a, other._b))

    def __ne__(self, other):
        """
        Compare self and other.

        EXAMPLES::

            sage: QuaternionAlgebra(-1,-7) != QuaternionAlgebra(-1,-7)
            False
            sage: QuaternionAlgebra(-1,-7) != QuaternionAlgebra(-1,-5)
            True
        """
        return not self.__eq__(other)

    def __hash__(self):
        """
        Compute the hash of ``self``.

        EXAMPLES::

            sage: h1 = hash(QuaternionAlgebra(-1,-7))
            sage: h2 = hash(QuaternionAlgebra(-1,-7))
            sage: h3 = hash(QuaternionAlgebra(-1,-5))
            sage: h1  == h2 and h1 != h3
            True
        """
        return hash((self.base_ring(), self._a, self._b))

    def gen(self, i=0):
        """
        Return the `i^{th}` generator of ``self``.

        INPUT:

        - ``i`` - integer (optional, default 0)

        EXAMPLES::

            sage: Q.<ii,jj,kk> = QuaternionAlgebra(QQ,-1,-2); Q
            Quaternion Algebra (-1, -2) with base ring Rational Field
            sage: Q.gen(0)
            ii
            sage: Q.gen(1)
            jj
            sage: Q.gen(2)
            kk
            sage: Q.gens()
            [ii, jj, kk]
        """
        return self._gens[i]

    def _repr_(self):
        """
        Print representation.

        TESTS::

            sage: Q.<i,j,k> = QuaternionAlgebra(QQ,-5,-2)
            sage: type(Q)
            <class 'sage.algebras.quatalg.quaternion_algebra.QuaternionAlgebra_ab_with_category'>
            sage: Q._repr_()
            'Quaternion Algebra (-5, -2) with base ring Rational Field'
            sage: Q
            Quaternion Algebra (-5, -2) with base ring Rational Field
            sage: print(Q)
            Quaternion Algebra (-5, -2) with base ring Rational Field
            sage: str(Q)
            'Quaternion Algebra (-5, -2) with base ring Rational Field'
        """
        return "Quaternion Algebra (%r, %r) with base ring %s" % (self._a, self._b, self.base_ring())

    def inner_product_matrix(self):
        """
        Return the inner product matrix associated to ``self``, i.e. the
        Gram matrix of the reduced norm as a quadratic form on ``self``.
        The standard basis `1`, `i`, `j`, `k` is orthogonal, so this matrix
        is just the diagonal matrix with diagonal entries `1`, `a`, `b`, `ab`.

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(-5,-19)
            sage: Q.inner_product_matrix()
            [  2   0   0   0]
            [  0  10   0   0]
            [  0   0  38   0]
            [  0   0   0 190]

            sage: R.<a,b> = QQ[]; Q.<i,j,k> = QuaternionAlgebra(Frac(R),a,b)
            sage: Q.inner_product_matrix()
            [    2     0     0     0]
            [    0  -2*a     0     0]
            [    0     0  -2*b     0]
            [    0     0     0 2*a*b]
        """
        a, b = self._a, self._b
        return diagonal_matrix(self.base_ring(), [2, -2*a, -2*b, 2*a*b])

    @cached_method
    def discriminant(self):
        """
        Given a quaternion algebra `A` defined over a number field,
        return the discriminant of `A`, i.e. the
        product of the ramified primes of `A`.

        EXAMPLES::

            sage: QuaternionAlgebra(210,-22).discriminant()
            210
            sage: QuaternionAlgebra(19).discriminant()
            19

            sage: F.<a> = NumberField(x^2-x-1)
            sage: B.<i,j,k> = QuaternionAlgebra(F, 2*a,F(-1))
            sage: B.discriminant()
            Fractional ideal (2)

            sage: QuaternionAlgebra(QQ[sqrt(2)],3,19).discriminant()
            Fractional ideal (1)
        """
        if not is_RationalField(self.base_ring()):
            try:
                F = self.base_ring()
                return F.hilbert_conductor(self._a, self._b)
            except NotImplementedError:
                raise ValueError("base field must be rational numbers or number field")
        else:
            return hilbert_conductor(self._a, self._b)

    def ramified_primes(self):
        """
        Return the primes that ramify in this quaternion algebra.

        Currently only implemented over the rational numbers.

        EXAMPLES::

            sage: QuaternionAlgebra(QQ, -1, -1).ramified_primes()
            [2]
        """
        #TODO: more examples

        return [f[0] for f in factor(self.discriminant())]

    def _magma_init_(self, magma):
        """
        Return Magma version of this quaternion algebra.

        EXAMPLES::

            sage: Q = QuaternionAlgebra(-1,-1); Q
            Quaternion Algebra (-1, -1) with base ring Rational Field
            sage: Q._magma_init_(magma)                                  # optional - magma
            'QuaternionAlgebra(_sage_[...],-1/1,-1/1)'
            sage: A = magma(Q); A                                        # optional - magma
            Quaternion Algebra with base ring Rational Field, defined by i^2 = -1, j^2 = -1
            sage: A.RamifiedPlaces()                                     # optional - magma
            [
            Ideal of Integer Ring generated by 2
            ]

        A more complicated example involving a quaternion algebra over a number field::

            sage: K.<a> = QQ[sqrt(2)]; Q = QuaternionAlgebra(K,-1,a); Q
            Quaternion Algebra (-1, sqrt2) with base ring Number Field in sqrt2 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?
            sage: magma(Q)                                              # optional - magma
            Quaternion Algebra with base ring Number Field with defining polynomial x^2 - 2 over the Rational Field, defined by i^2 = -1, j^2 = sqrt2
            sage: Q._magma_init_(magma)                                 # optional - magma
            'QuaternionAlgebra(_sage_[...],(_sage_[...]![-1, 0]),(_sage_[...]![0, 1]))'
        """
        R = magma(self.base_ring())
        return 'QuaternionAlgebra(%s,%s,%s)' % (R.name(),
                                                self._a._magma_init_(magma),
                                                self._b._magma_init_(magma))

    def quaternion_order(self, basis, check=True):
        """
        Return the order of this quaternion order with given basis.

        INPUT:

        - ``basis`` - list of 4 elements of ``self``
        - ``check`` - bool (default: ``True``)

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(-11,-1)
            sage: Q.quaternion_order([1,i,j,k])
            Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis (1, i, j, k)

        We test out ``check=False``::

            sage: Q.quaternion_order([1,i,j,k], check=False)
            Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis [1, i, j, k]
            sage: Q.quaternion_order([i,j,k], check=False)
            Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis [i, j, k]
        """
        return QuaternionOrder(self, basis, check=check)

    def ideal(self, gens, left_order=None, right_order=None, check=True, **kwds):
        r"""
        Return the quaternion ideal with given gens over `\ZZ`.

        Neither a left or right order structure need be specified.

        INPUT:

        - ``gens`` -- a list of elements of this quaternion order

        - ``check`` -- bool (default: ``True``); if ``False``, then ``gens`` must
          4-tuple that forms a Hermite basis for an ideal

        - ``left_order`` -- a quaternion order or ``None``

        - ``right_order`` -- a quaternion order or ``None``

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1)
            sage: R.ideal([2*a for a in R.basis()])
            Fractional ideal (2, 2*i, 2*j, 2*k)
        """
        if self.base_ring() == QQ:
            return QuaternionFractionalIdeal_rational(gens, left_order=left_order, right_order=right_order, check=check)
        else:
            raise NotImplementedError("ideal only implemented for quaternion algebras over QQ")

    @cached_method
    def modp_splitting_data(self, p):
        r"""
        Return mod `p` splitting data for this quaternion algebra at
        the unramified prime `p`.

        This is `2\times 2`
        matrices `I`, `J`, `K` over the finite field `\GF{p}` such that if
        the quaternion algebra has generators `i, j, k`, then `I^2 =
        i^2`, `J^2 = j^2`, `IJ=K` and `IJ=-JI`.

        .. NOTE::

           Currently only implemented when `p` is odd and the base
           ring is `\QQ`.

        INPUT:

        - `p` -- unramified odd prime

        OUTPUT:

        - 2-tuple of matrices over finite field

        EXAMPLES::

            sage: Q = QuaternionAlgebra(-15, -19)
            sage: Q.modp_splitting_data(7)
            (
            [0 6]  [6 1]  [6 6]
            [1 0], [1 1], [6 1]
            )
            sage: Q.modp_splitting_data(next_prime(10^5))
            (
            [    0 99988]  [97311     4]  [99999 59623]
            [    1     0], [13334  2692], [97311     4]
            )
            sage: I,J,K = Q.modp_splitting_data(23)
            sage: I
            [0 8]
            [1 0]
            sage: I^2
            [8 0]
            [0 8]
            sage: J
            [19  2]
            [17  4]
            sage: J^2
            [4 0]
            [0 4]
            sage: I*J == -J*I
            True
            sage: I*J == K
            True

        The following is a good test because of the asserts in the code::

            sage: v = [Q.modp_splitting_data(p) for p in primes(20,1000)]

        Proper error handling::

            sage: Q.modp_splitting_data(5)
            Traceback (most recent call last):
            ...
            NotImplementedError: algorithm for computing local splittings not implemented in general (currently require the first invariant to be coprime to p)

            sage: Q.modp_splitting_data(2)
            Traceback (most recent call last):
            ...
            NotImplementedError: p must be odd
        """
        if self.base_ring() != QQ:
            raise NotImplementedError("must be rational quaternion algebra")
        p = ZZ(p)
        if not p.is_prime():
            raise ValueError("p (=%s) must be prime" % p)
        if p == 2:
            raise NotImplementedError("p must be odd")
        if self.discriminant() % p == 0:
            raise ValueError("p (=%s) must be an unramified prime" % p)

        i, j, k = self.gens()
        F = GF(p)
        i2 = F(i*i)
        j2 = F(j*j)

        M = MatrixSpace(F, 2)
        I = M([0,i2,1,0])
        if i2 == 0:
            raise NotImplementedError("algorithm for computing local splittings not implemented in general (currently require the first invariant to be coprime to p)")
        i2inv = 1/i2
        a = None
        for b in list(F):
            if not b: continue
            c = j2 + i2inv * b*b
            if c.is_square():
                a = -c.sqrt()
                break

        if a is None:
            # do a fallback search, maybe needed in char 3 sometimes.
            for J in M:
                K = I*J
                if J*J == j2 and K == -J*I:
                    return I, J, K

        J = M([a,b,(j2-a*a)/b, -a])
        K = I*J
        assert K == -J*I, "bug in that I,J don't skew commute"
        return I, J, K

    def modp_splitting_map(self, p):
        r"""
        Return Python map from the (`p`-integral) quaternion algebra to
        the set of `2\times 2` matrices over `\GF{p}`.

        INPUT:

        - `p` -- prime number

        EXAMPLES::

            sage: Q.<i,j,k> = QuaternionAlgebra(-1, -7)
            sage: f = Q.modp_splitting_map(13)
            sage: a = 2+i-j+3*k; b = 7+2*i-4*j+k
            sage: f(a*b)
            [12  3]
            [10  5]
            sage: f(a)*f(b)
            [12  3]
            [10  5]
        """
        I, J, K = self.modp_splitting_data(p)
        F = I.base_ring()
        def phi(q):
            v = [F(a) for a in q.coefficient_tuple()]
            return v[0] + I*v[1] + J*v[2] + K*v[3]
        return phi


############################################################
# Unpickling
############################################################
def unpickle_QuaternionAlgebra_v0(*key):
    """
    The 0th version of pickling for quaternion algebras.

    EXAMPLES::

        sage: Q = QuaternionAlgebra(-5,-19)
        sage: t = (QQ, -5, -19, ('i', 'j', 'k'))
        sage: sage.algebras.quatalg.quaternion_algebra.unpickle_QuaternionAlgebra_v0(*t)
        Quaternion Algebra (-5, -19) with base ring Rational Field
        sage: loads(dumps(Q)) == Q
        True
        sage: loads(dumps(Q)) is Q
        True
    """
    return QuaternionAlgebra(*key)


class QuaternionOrder(Algebra):
    """
    An order in a quaternion algebra.

    EXAMPLES::

        sage: QuaternionAlgebra(-1,-7).maximal_order()
        Order of Quaternion Algebra (-1, -7) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)
        sage: type(QuaternionAlgebra(-1,-7).maximal_order())
        <class 'sage.algebras.quatalg.quaternion_algebra.QuaternionOrder_with_category'>
    """
    def __init__(self, A, basis, check=True):
        """
        INPUT:

        - ``A`` - a quaternion algebra
        - ``basis`` - list of 4 integral quaternions in ``A``
        - ``check`` - whether to do type and other consistency checks

        .. WARNING::

            Currently most methods silently assume that the ``A.base_ring()``
            is ``QQ``.

        EXAMPLES::

            sage: A.<i,j,k> = QuaternionAlgebra(-3,-5)
            sage: sage.algebras.quatalg.quaternion_algebra.QuaternionOrder(A, [1,i,j,k])
            Order of Quaternion Algebra (-3, -5) with base ring Rational Field with basis (1, i, j, k)
            sage: R = sage.algebras.quatalg.quaternion_algebra.QuaternionOrder(A, [1,2*i,2*j,2*k]); R
            Order of Quaternion Algebra (-3, -5) with base ring Rational Field with basis (1, 2*i, 2*j, 2*k)
            sage: type(R)
            <class 'sage.algebras.quatalg.quaternion_algebra.QuaternionOrder_with_category'>

            Over QQ and number fields it is checked whether the given
            basis actually gives a an order (as a module over the maximal order):

            sage: A.<i,j,k> = QuaternionAlgebra(-1,-1)
            sage: A.quaternion_order([1,i,j,i-j])
            Traceback (most recent call last):
            ...
            ValueError: basis must have rank 4
            sage: A.quaternion_order([2,i,j,k])
            Traceback (most recent call last):
            ...
            ValueError: lattice must contain 1
            sage: A.quaternion_order([1,i/2,j/2,k/2])
            Traceback (most recent call last):
            ...
            ValueError: given lattice must be a ring

            sage: K = QuadraticField(10)
            sage: A.<i,j,k> = QuaternionAlgebra(K,-1,-1)
            sage: A.quaternion_order([1,i,j,k])
            Order of Quaternion Algebra (-1, -1) with base ring Number Field in a with defining polynomial x^2 - 10 with a = 3.162277660168380? with basis (1, i, j, k)
            sage: A.quaternion_order([1,i/2,j,k])
            Traceback (most recent call last):
            ...
            ValueError: given lattice must be a ring

        TESTS::

            sage: TestSuite(R).run()
        """
        if check:
            # right data type
            if not isinstance(basis, (list, tuple)):
                raise TypeError("basis must be a list or tuple")
            # right length
            if len(basis) != 4:
                raise ValueError("basis must have length 4")
            # coerce to common parent
            basis = tuple([A(x) for x in basis])

            # has rank 4
            V = A.base_ring()**4
            if V.span([ V(x.coefficient_tuple()) for x in basis]).dimension() != 4:
                raise ValueError("basis must have rank 4")

            # The additional checks will work over QQ and over number fields,
            # but we can't actually do much with an order defined over a number
            # field

            if A.base_ring() == QQ:     # fast code over QQ
                M = matrix(QQ, 4, 4, [ x.coefficient_tuple() for x in basis])
                v = M.solve_left(V([1,0,0,0]))

                if v.denominator() != 1:
                    raise ValueError("lattice must contain 1")

                # check if multiplicatively closed
                M1 = basis_for_quaternion_lattice(basis)
                M2 = basis_for_quaternion_lattice(list(basis) + [ x*y for x in basis for y in basis])
                if M1 != M2:
                    raise ValueError("given lattice must be a ring")

            if A.base_ring() != QQ:     # slow code over number fields (should eventually use PARI's nfhnf)
                O = None
                try:
                    O = A.base_ring().maximal_order()
                except AttributeError:
                    pass

                if O:
                    M = matrix(A.base_ring(), 4, 4, [ x.coefficient_tuple() for x in basis])
                    v = M.solve_left(V([1,0,0,0]))

                    if any(a not in O for a in v):
                        raise ValueError("lattice must contain 1")

                    # check if multiplicatively closed
                    Y = matrix(QQ, 16, 4, [ (x*y).coefficient_tuple() for x in basis for y in basis])
                    X = M.solve_left(Y)
                    if any(a not in O for x in X for a in x):
                        raise ValueError("given lattice must be a ring")

        self.__basis = basis
        self.__quaternion_algebra = A
        Parent.__init__(self, base=ZZ, facade=(A,), category=Algebras(ZZ))

    def gens(self):
        """
        Return generators for self.

        EXAMPLES::

            sage: QuaternionAlgebra(-1,-7).maximal_order().gens()
            (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)
        """
        return self.__basis

    def ngens(self):
        """
        Return the number of generators (which is 4).

        EXAMPLES::

            sage: QuaternionAlgebra(-1,-7).maximal_order().ngens()
            4
        """
        return 4

    def gen(self, n):
        """
        Return the n-th generator.

        INPUT:

        - ``n`` - an integer between 0 and 3, inclusive.

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order(); R
            Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
            sage: R.gen(0)
            1/2 + 1/2*i
            sage: R.gen(1)
            1/2*j - 1/2*k
            sage: R.gen(2)
            i
            sage: R.gen(3)
            -k
        """
        return self.__basis[n]

    def __eq__(self, R):
        """
        Compare orders self and other.

        Two orders are equal if they
        have the same basis and are in the same quaternion algebra.

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R == R                       # indirect doctest
            True
            sage: R == QuaternionAlgebra(-1,-1).maximal_order()
            False
            sage: R == 5
            False
        """
        if not isinstance(R, QuaternionOrder):
            return False
        return (self.__quaternion_algebra == R.__quaternion_algebra and
                self.__basis == R.__basis)

    def __ne__(self, other):
        """
        Compare orders self and other.

        Two orders are equal if they
        have the same basis and are in the same quaternion algebra.

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R != R                       # indirect doctest
            False
            sage: R != QuaternionAlgebra(-1,-1).maximal_order()
            True
        """
        return not self.__eq__(other)

    def __hash__(self):
        """
        Compute the hash of ``self``.

        EXAMPLES::

            sage: h1 = hash(QuaternionAlgebra(-1,-7).maximal_order())
            sage: h2 = hash(QuaternionAlgebra(-1,-7).maximal_order())
            sage: h3 = hash(QuaternionAlgebra(-1,-5).maximal_order())
            sage: h1  == h2 and h1 != h3
            True
        """
        return hash((self.__quaternion_algebra, self.__basis))

    def basis(self):
        """
        Return fix choice of basis for this quaternion order.

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().basis()
            (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
        """
        return self.__basis

    def quaternion_algebra(self):
        """
        Return ambient quaternion algebra that contains this quaternion order.

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().quaternion_algebra()
            Quaternion Algebra (-11, -1) with base ring Rational Field
        """
        return self.__quaternion_algebra

    def _repr_(self):
        """
        Return string representation of this order.

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order()._repr_()
            'Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)'
            sage: QuaternionAlgebra(-11,-1).maximal_order()
            Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
        """
        return 'Order of %s with basis %s' % (self.quaternion_algebra(), self.basis())

    def random_element(self, *args, **kwds):
        """
        Return a random element of this order.

        The args and kwds are passed to the random_element method of
        the integer ring, and we return an element of the form

        .. MATH::

            ae_1 + be_2 + ce_3 + de_4

        where `e_1`, ..., `e_4` are the basis of this order and `a`,
        `b`, `c`, `d` are random integers.

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().random_element()  # random
            -4 - 4*i + j - k
            sage: QuaternionAlgebra(-11,-1).maximal_order().random_element(-10,10)  # random
            -9/2 - 7/2*i - 7/2*j - 3/2*k
        """
        return sum(ZZ.random_element(*args, **kwds) * b for b in self.basis())

    def intersection(self, other):
        """
        Return the intersection of this order with other.

        INPUT:

        - ``other`` - a quaternion order in the same ambient quaternion algebra

        OUTPUT: a quaternion order

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.intersection(R)
            Order of Quaternion Algebra (-11, -1) with base ring Rational Field with basis (1/2 + 1/2*i, i, 1/2*j + 1/2*k, k)

        We intersect various orders in the quaternion algebra ramified at 11::

            sage: B = BrandtModule(11,3)
            sage: R = B.maximal_order(); S = B.order_of_level_N()
            sage: R.intersection(S)
            Order of Quaternion Algebra (-1, -11) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 5/2*k, j, 3*k)
            sage: R.intersection(S) == S
            True
            sage: B = BrandtModule(11,5)
            sage: T = B.order_of_level_N()
            sage: S.intersection(T)
            Order of Quaternion Algebra (-1, -11) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 23/2*k, j, 15*k)
        """
        if not isinstance(other, QuaternionOrder):
            raise TypeError("other must be a QuaternionOrder")

        A = self.quaternion_algebra()
        if other.quaternion_algebra() != A:
            raise ValueError("self and other must be in the same ambient quaternion algebra")

        V = A.base_ring()**4

        B = V.span([V(list(g)) for g in self.basis()], ZZ)
        C = V.span([V(list(g)) for g in other.basis()], ZZ)

        # todo -- A(list(e)) could be A(e)
        return QuaternionOrder(A, [A(list(e)) for e in B.intersection(C).basis()])

    @cached_method
    def free_module(self):
        r"""
        Return the free `\ZZ`-module that corresponds to this order
        inside the vector space corresponding to the ambient
        quaternion algebra.

        OUTPUT:

        A free `\ZZ`-module of rank 4.

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.basis()
            (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
            sage: R.free_module()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [1/2 1/2   0   0]
            [  0   1   0   0]
            [  0   0 1/2 1/2]
            [  0   0   0   1]
        """
        V = self.quaternion_algebra().base_ring()**4
        return V.span([V(list(g)) for g in self.basis()], ZZ)

    def discriminant(self):
        r"""
        Return the discriminant of this order.

        This is defined as
        `\sqrt{ det ( Tr(e_i \bar{e}_j ) ) }`, where `\{e_i\}` is the
        basis of the order.

        OUTPUT: rational number

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().discriminant()
            11
            sage: S = BrandtModule(11,5).order_of_level_N()
            sage: S.discriminant()
            55
            sage: type(S.discriminant())
            <... 'sage.rings.rational.Rational'>
        """
        L = []
        for d in self.basis():
            MM = []
            for e in self.basis():
                MM.append((d * e.conjugate()).reduced_trace())
            L.append(MM)

        return (MatrixSpace(QQ, 4, 4)(L)).determinant().sqrt()

    def left_ideal(self, gens, check=True):
        r"""
        Return the ideal with given gens over `\ZZ`.

        INPUT:

        - ``gens`` -- a list of elements of this quaternion order

        - ``check`` -- bool (default: ``True``); if ``False``, then ``gens`` must
          4-tuple that forms a Hermite basis for an ideal

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.left_ideal([2*a for a in R.basis()])
            Fractional ideal (1 + i, 2*i, j + k, 2*k)
        """
        if self.base_ring() == ZZ:
            return QuaternionFractionalIdeal_rational(gens, left_order=self, check=check)
        else:
            raise NotImplementedError("ideal only implemented for quaternion algebras over QQ")

    def right_ideal(self, gens, check=True):
        r"""
        Return the ideal with given gens over `\ZZ`.

        INPUT:

        - ``gens`` -- a list of elements of this quaternion order

        - ``check`` -- bool (default: ``True``); if ``False``, then ``gens`` must
          4-tuple that forms a Hermite basis for an ideal

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.right_ideal([2*a for a in R.basis()])
            Fractional ideal (1 + i, 2*i, j + k, 2*k)
        """
        if self.base_ring() == ZZ:
            return QuaternionFractionalIdeal_rational(gens, right_order=self, check=check)
        else:
            raise NotImplementedError("ideal only implemented for quaternion algebras over QQ")

    def unit_ideal(self):
        """
        Return the unit ideal in this quaternion order.

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: I = R.unit_ideal(); I
            Fractional ideal (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
        """
        if self.base_ring() == ZZ:
            return QuaternionFractionalIdeal_rational(self.basis(), left_order=self, right_order=self, check=False)
        else:
            raise NotImplementedError("ideal only implemented for quaternion algebras over QQ")

    def quadratic_form(self):
        """
        Return the normalized quadratic form associated to this quaternion order.

        OUTPUT: quadratic form

        EXAMPLES::

            sage: R = BrandtModule(11,13).order_of_level_N()
            sage: Q = R.quadratic_form(); Q
            Quadratic form in 4 variables over Rational Field with coefficients:
            [ 14 253 55 286 ]
            [ * 1455 506 3289 ]
            [ * * 55 572 ]
            [ * * * 1859 ]
            sage: Q.theta_series(10)
            1 + 2*q + 2*q^4 + 4*q^6 + 4*q^8 + 2*q^9 + O(q^10)
        """
        return self.unit_ideal().quadratic_form()

    def ternary_quadratic_form(self, include_basis=False):
        r"""
        Return the ternary quadratic form associated to this order.

        INPUT:

        - ``include_basis`` -- bool (default: False), if True also
          return a basis for the dimension 3 subspace `G`

        OUTPUT:

        - QuadraticForm

        - optional basis for dimension 3 subspace

        This function computes the positive definition quadratic form
        obtained by letting G be the trace zero subspace of `\ZZ` +
        2* ``self``, which has rank 3, and restricting the pairing::

           (x,y) = (x.conjugate()*y).reduced_trace()

        to `G`.

        APPLICATIONS: Ternary quadratic forms associated to an order
        in a rational quaternion algebra are useful in computing with
        Gross points, in decided whether quaternion orders have
        embeddings from orders in quadratic imaginary fields, and in
        computing elements of the Kohnen plus subspace of modular
        forms of weight 3/2.

        EXAMPLES::

            sage: R = BrandtModule(11,13).order_of_level_N()
            sage: Q = R.ternary_quadratic_form(); Q
            Quadratic form in 3 variables over Rational Field with coefficients:
            [ 5820 1012 13156 ]
            [ * 55 1144 ]
            [ * * 7436 ]
            sage: factor(Q.disc())
            2^4 * 11^2 * 13^2

        The following theta series is a modular form of weight 3/2 and level 4*11*13::

            sage: Q.theta_series(100)
            1 + 2*q^23 + 2*q^55 + 2*q^56 + 2*q^75 + 4*q^92 + O(q^100)
        """
        if self.base_ring() != ZZ:
            raise NotImplementedError("ternary quadratic form of order only implemented for quaternion algebras over QQ")

        Q = self.quaternion_algebra()
        # 2*R + ZZ
        twoR = self.free_module().scale(2)
        Z = twoR.span([Q(1).coefficient_tuple()], ZZ)
        S = twoR + Z
        # Now we intersect with the trace 0 submodule
        v = [b.reduced_trace() for b in Q.basis()]
        M = matrix(QQ,4,1,v)
        tr0 = M.kernel()
        G = tr0.intersection(S)
        B = [Q(a) for a in G.basis()]
        m = matrix(QQ,[[x.pair(y) for x in B] for y in B])
        from sage.quadratic_forms.quadratic_form import QuadraticForm
        Q = QuadraticForm(m)
        if include_basis:
            return Q, B
        else:
            return Q


class QuaternionFractionalIdeal(Ideal_fractional):
    pass


class QuaternionFractionalIdeal_rational(QuaternionFractionalIdeal):
    """
    A fractional ideal in a rational quaternion algebra.

    INPUT:

    - ``left_order`` -- a quaternion order or ``None``

    - ``right_order`` -- a quaternion order or ``None``

    - ``basis`` -- tuple of length 4 of elements in of ambient
      quaternion algebra whose `\\ZZ`-span is an ideal

    - ``check`` -- bool (default: ``True``); if ``False``, do no type
      checking, and the input basis *must* be in Hermite form.
    """
    def __init__(self, basis, left_order=None, right_order=None, check=True):
        """
        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.right_ideal(R.basis())
            Fractional ideal (1/2 + 1/2*i, i, 1/2*j + 1/2*k, k)
            sage: R.right_ideal(tuple(R.basis()), check=False)
            Fractional ideal (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
        """
        if check:
            if left_order is not None and not isinstance(left_order, QuaternionOrder):
                raise TypeError("left_order must be a quaternion order or None")
            if right_order is not None and not isinstance(right_order, QuaternionOrder):
                raise TypeError("right_order must be a quaternion order or None")
            if not isinstance(basis, (list, tuple)):
                raise TypeError("basis must be a list or tuple")

        self.__left_order = left_order
        self.__right_order = right_order

        if check:
            try:
                Q = self.quaternion_order().quaternion_algebra()
            except RuntimeError:
                Q = basis[0].parent()
            basis = tuple([Q(v) for v in
                           (QQ**4).span([Q(v).coefficient_tuple() for v in basis], ZZ).basis()])
        self.__basis = basis

    def scale(self, alpha, left=False):
        r"""
        Scale the fractional ideal ``self`` by multiplying the basis
        by ``alpha``.

        INPUT:

        - `\alpha` -- element of quaternion algebra

        - ``left`` -- bool (default: False); if true multiply
          `\alpha` on the left, otherwise multiply `\alpha` on the right

        OUTPUT:

        - a new fractional ideal

        EXAMPLES::

            sage: B = BrandtModule(5,37); I = B.right_ideals()[0]; i,j,k = B.quaternion_algebra().gens(); I
            Fractional ideal (2 + 2*j + 106*k, i + 2*j + 105*k, 4*j + 64*k, 148*k)
            sage: I.scale(i)
            Fractional ideal [2*i + 212*j - 2*k, -2 + 210*j - 2*k, 128*j - 4*k, 296*j]
            sage: I.scale(i, left=True)
            Fractional ideal [2*i - 212*j + 2*k, -2 - 210*j + 2*k, -128*j + 4*k, -296*j]
            sage: I.scale(i, left=False)
            Fractional ideal [2*i + 212*j - 2*k, -2 + 210*j - 2*k, 128*j - 4*k, 296*j]
            sage: i * I.gens()[0]
            2*i - 212*j + 2*k
            sage: I.gens()[0] * i
            2*i + 212*j - 2*k
        """
        Q = self.quaternion_algebra()
        alpha = Q(alpha)
        if left:
            gens = [alpha * b for b in self.basis()]
        else:
            gens = [b * alpha for b in self.basis()]
        return Q.ideal(gens, left_order=self.__left_order,
                       right_order=self.__right_order, check=False)

    def quaternion_algebra(self):
        """
        Return the ambient quaternion algebra that contains this fractional ideal.

        OUTPUT: a quaternion algebra

        EXAMPLES::

            sage: I = BrandtModule(3,5).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 34*k, 8*j + 32*k, 40*k)
            sage: I.quaternion_algebra()
            Quaternion Algebra (-1, -3) with base ring Rational Field
        """
        try:
            return self.__quaternion_algebra
        except AttributeError:
            pass
        A = self.__basis[0].parent()
        self.__quaternion_algebra = A
        return A

    def _compute_order(self, side='left'):
        r"""
        Used internally to compute either the left or right order
        associated to an ideal in a quaternion algebra.  If
        action='right', compute the left order, and if action='left'
        compute the right order.

        INPUT:

        - ``side`` -- 'left' or 'right'

        EXAMPLES::

            sage: R.<i,j,k> = QuaternionAlgebra(-1,-11)
            sage: I = R.ideal([2 + 2*j + 140*k, 2*i + 4*j + 150*k, 8*j + 104*k, 152*k])
            sage: Ol = I._compute_order('left'); Ol
            Order of Quaternion Algebra (-1, -11) with base ring Rational Field with basis (1/2 + 1/2*j + 35*k, 1/4*i + 1/2*j + 75/4*k, j + 32*k, 38*k)
            sage: Or = I._compute_order('right'); Or
            Order of Quaternion Algebra (-1, -11) with base ring Rational Field with basis (1/2 + 1/2*j + 16*k, 1/2*i + 11/2*k, j + 13*k, 19*k)
            sage: Ol.discriminant()
            209
            sage: Or.discriminant()
            209
            sage: I.left_order() == Ol
            True
            sage: I.right_order() == Or
            True

        ALGORITHM: Let `b_1, b_2, b_3, b_3` be a basis for this
        fractional ideal `I`, and assume we want to compute the left
        order of `I` in the quaternion algebra `Q`.  Then
        multiplication by `b_i` on the right defines a map `B_i:Q \to
        Q`.  We have

        .. MATH::

           R = B_1^{-1}(I) \cap B_2^{-1}(I) \cap B_3^{-1}(I)\cap B_4^{-1}(I).

        This is because

        .. MATH::

           B_n^{-1}(I) = \{\alpha \in Q : \alpha b_n \in I \},

        and

        .. MATH::

           R = \{\alpha \in Q : \alpha b_n \in I, n=1,2,3,4\}.
        """
        if side == 'left':
            action = 'right'
        elif side == 'right':
            action = 'left'
        else:
            raise ValueError("side must be 'left' or 'right'")
        Q = self.quaternion_algebra()
        if Q.base_ring() != QQ:
            raise NotImplementedError("computation of left and right orders only implemented over QQ")
        M = [(~b).matrix(action=action) for b in self.basis()]
        B = self.basis_matrix()
        invs = [B * m for m in M]
        # Now intersect the row spans of each matrix in invs
        ISB = [Q(v) for v in intersection_of_row_modules_over_ZZ(invs).row_module(ZZ).basis()]
        return Q.quaternion_order(ISB)

    def left_order(self):
        """
        Return the left order associated to this fractional ideal.

        OUTPUT: an order in a quaternion algebra

        EXAMPLES::

            sage: B = BrandtModule(11)
            sage: R = B.maximal_order()
            sage: I = R.unit_ideal()
            sage: I.left_order()
            Order of Quaternion Algebra (-1, -11) with base ring Rational Field with basis (1/2 + 1/2*j, 1/2*i + 1/2*k, j, k)

        We do a consistency check::

            sage: B = BrandtModule(11,19); R = B.right_ideals()
            sage: [r.left_order().discriminant() for r in R]
            [209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209, 209]
        """
        if self.__left_order is None:
            self.__left_order = self._compute_order(side='left')
        return self.__left_order

    def right_order(self):
        """
        Return the right order associated to this fractional ideal.

        OUTPUT: an order in a quaternion algebra

        EXAMPLES::

            sage: I = BrandtModule(389).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 2*k, i + 2*j + k, 8*j, 8*k)
            sage: I.right_order()
            Order of Quaternion Algebra (-2, -389) with base ring Rational Field with basis (1/2 + 1/2*j + 1/2*k, 1/4*i + 1/2*j + 1/4*k, j, k)
            sage: I.left_order()
            Order of Quaternion Algebra (-2, -389) with base ring Rational Field with basis (1/2 + 1/2*j + 3/2*k, 1/8*i + 1/4*j + 9/8*k, j + k, 2*k)

        The following is a big consistency check.  We take reps for
        all the right ideal classes of a certain order, take the
        corresponding left orders, then take ideals in the left orders
        and from those compute the right order again::

            sage: B = BrandtModule(11,19); R = B.right_ideals()
            sage: O = [r.left_order() for r in R]
            sage: J = [O[i].left_ideal(R[i].basis()) for i in range(len(R))]
            sage: len(set(J))
            18
            sage: len(set([I.right_order() for I in J]))
            1
            sage: J[0].right_order() == B.order_of_level_N()
            True
        """
        if self.__right_order is None:
            self.__right_order = self._compute_order(side='right')
        return self.__right_order

    def __repr__(self):
        """
        Return string representation of this quaternion fractional ideal.

        EXAMPLES::

            sage: I = BrandtModule(11).right_ideals()[1]
            sage: type(I)
            <class 'sage.algebras.quatalg.quaternion_algebra.QuaternionFractionalIdeal_rational'>
            sage: I.__repr__()
            'Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 2*k, 8*j, 8*k)'
        """
        return 'Fractional ideal %s' % (self.gens(),)

    def quaternion_order(self):
        """
        Return the order for which this ideal is a left or right
        fractional ideal.

        If this ideal has both a left and right
        ideal structure, then the left order is returned.  If it has
        neither structure, then an error is raised.

        OUTPUT: QuaternionOrder

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.unit_ideal().quaternion_order() is R
            True
        """
        try: return self.__quaternion_order
        except AttributeError: pass
        if self.__left_order is not None:
            A = self.__left_order
        elif self.__right_order is not None:
            A = self.__right_order
        else:
            raise RuntimeError("unable to determine quaternion order of ideal without known order")
        self.__quaternion_order = A
        return A

    def ring(self):
        """
        Return ring that this is a fractional ideal for.

        EXAMPLES::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: R.unit_ideal().ring() is R
            True
        """
        return self.quaternion_order()

    def basis(self):
        """
        Return basis for this fractional ideal.

        The basis is in Hermite form.

        OUTPUT: tuple

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().unit_ideal().basis()
            (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
        """
        return self.__basis

    def gens(self):
        r"""
        Return the generators for this ideal, which are the same as
        the `\ZZ`-basis for this ideal.

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().unit_ideal().gens()
            (1/2 + 1/2*i, 1/2*j - 1/2*k, i, -k)
        """
        return self.__basis

    def __eq__(self, right):
        """
        Compare this fractional quaternion ideal to ``right``.

        If ``right`` is not a fractional quaternion ideal, return ``False``.

        If the fractional ideals are in different ambient
        quaternion algebras, then the quaternion algebras themselves
        are compared.

        INPUT:

        - ``right`` - another fractional quaternion ideal

        EXAMPLES::

            sage: I = QuaternionAlgebra(-11,-1).maximal_order().unit_ideal()
            sage: I == I                # indirect doctest
            True
            sage: I == 5
            False
        """
        if not isinstance(right, QuaternionFractionalIdeal_rational):
            return False
        return self.__basis == right.__basis

    def __ne__(self, other):
        """
        Compare this fractional quaternion ideal to ``right``.

        INPUT:

        - ``right`` - another fractional quaternion ideal

        EXAMPLES::

            sage: I = QuaternionAlgebra(-11,-1).maximal_order().unit_ideal()
            sage: I != I                # indirect doctest
            False
        """
        return not self.__eq__(other)

    def __hash__(self):
        """
        Return the hash of ``self``.

        EXAMPLES::

            sage: I = QuaternionAlgebra(-11,-1).maximal_order().unit_ideal()
            sage: hash(I) == hash(I)
            True

        TESTS::

            sage: R = QuaternionAlgebra(-11,-1).maximal_order()
            sage: H = hash(R.right_ideal(R.basis()))
        """
        return hash(self.__basis)

    def basis_matrix(self):
        r"""
        Return basis matrix `M` in Hermite normal form for self as a
        matrix with rational entries.

        If `Q` is the ambient quaternion algebra, then the `\ZZ`-span of
        the rows of `M` viewed as linear combinations of Q.basis() =
        `[1,i,j,k]` is the fractional ideal self.  Also,
        ``M * M.denominator()`` is an integer matrix in Hermite normal form.

        OUTPUT: matrix over `\QQ`

        EXAMPLES::

            sage: QuaternionAlgebra(-11,-1).maximal_order().unit_ideal().basis_matrix()
            [ 1/2  1/2    0    0]
            [   0    0  1/2 -1/2]
            [   0    1    0    0]
            [   0    0    0   -1]
        """
        try: return self.__hermite_basis_matrix
        except AttributeError: pass
        B = quaternion_algebra_cython.rational_matrix_from_rational_quaternions(self.__basis)
        self.__hermite_basis_matrix = B
        return B

    def theta_series_vector(self, B):
        r"""
        Return theta series coefficients of ``self``, as a vector
        of ``B`` integers.

        INPUT:

        - ``B`` -- positive integer

        OUTPUT:

        Vector over `\ZZ` with ``B`` entries.

        EXAMPLES::

            sage: I = BrandtModule(37).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 2*k, i + 2*j + k, 8*j, 8*k)
            sage: I.theta_series_vector(5)
            (1, 0, 2, 2, 6)
            sage: I.theta_series_vector(10)
            (1, 0, 2, 2, 6, 4, 8, 6, 10, 10)
            sage: I.theta_series_vector(5)
            (1, 0, 2, 2, 6)
        """
        B = Integer(B)
        try:
            if len(self.__theta_series_vector)>= B:
                return self.__theta_series_vector[:B]
        except AttributeError:
            pass
        V = FreeModule(ZZ, B)
        Q = self.quadratic_form()
        v = V(Q.representation_number_list(B))
        self.__theta_series_vector = v
        return v

    @cached_method
    def quadratic_form(self):
        """
        Return the normalized quadratic form associated to this quaternion ideal.

        OUTPUT: quadratic form

        EXAMPLES::

            sage: I = BrandtModule(11).right_ideals()[1]
            sage: Q = I.quadratic_form(); Q
            Quadratic form in 4 variables over Rational Field with coefficients:
            [ 18 22 33 22 ]
            [ * 7 22 11 ]
            [ * * 22 0 ]
            [ * * * 22 ]
            sage: Q.theta_series(10)
            1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + 36*q^8 + 36*q^9 + O(q^10)
            sage: I.theta_series(10)
            1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + 36*q^8 + 36*q^9 + O(q^10)
        """
        from sage.quadratic_forms.quadratic_form import QuadraticForm
        # first get the gram matrix
        gram_matrix = self.gram_matrix()
        # rescale so that there are no denominators
        gram_matrix, _ = gram_matrix._clear_denom()
        # Make sure gcd of all entries is 1.
        g = gram_matrix.gcd()
        if g != 1:
            gram_matrix = gram_matrix / g
        # now get the quadratic form
        return QuadraticForm(gram_matrix)

    def theta_series(self, B, var='q'):
        r"""
        Return normalized theta series of ``self``, as a power series over
        `\ZZ` in the variable ``var``, which is 'q' by default.

        The normalized theta series is by definition

        .. MATH::

            \theta_I(q) = \sum_{x \in I} q^{\frac{N(x)}{N(I)}}.

        INPUT:

        - ``B`` -- positive integer
        - ``var`` -- string (default: 'q')

        OUTPUT: power series

        EXAMPLES::

            sage: I = BrandtModule(11).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 2*k, 8*j, 8*k)
            sage: I.norm()
            32
            sage: I.theta_series(5)
            1 + 12*q^2 + 12*q^3 + 12*q^4 + O(q^5)
            sage: I.theta_series(5,'T')
            1 + 12*T^2 + 12*T^3 + 12*T^4 + O(T^5)
            sage: I.theta_series(3)
            1 + 12*q^2 + O(q^3)
        """
        try:
            if self.__theta_series.prec() >= B:
                if var == self.__theta_series.variable():
                    return self.__theta_series.add_bigoh(B)
                else:
                    p_ring = self._theta_series.parent().change_var(var)
                    p_ring(self.__theta_series.list()[:B+1])
        except AttributeError:
            pass
        v = self.theta_series_vector(B)
        p_ring = PowerSeriesRing(ZZ, var)
        theta = p_ring(v.list()).add_bigoh(B)
        self.__theta_series = theta
        return theta

    @cached_method
    def gram_matrix(self):
        r"""
        Return the Gram matrix of this fractional ideal.

        OUTPUT: `4 \times 4` matrix over `\QQ`.

        EXAMPLES::

            sage: I = BrandtModule(3,5).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 34*k, 8*j + 32*k, 40*k)
            sage: I.gram_matrix()
            [  640  1920  2112  1920]
            [ 1920 14080 13440 16320]
            [ 2112 13440 13056 15360]
            [ 1920 16320 15360 19200]
        """
        A = self.__basis
        B = [z.conjugate() for z in self.__basis]
        two = QQ(2)
        m = [two * (a * b).reduced_trace() for b in B for a in A]
        M44 = MatrixSpace(QQ, 4, 4)
        return M44(m, coerce=False)

    def norm(self):
        """
        Return the reduced norm of this fractional ideal.

        OUTPUT: rational number

        EXAMPLES::

            sage: M = BrandtModule(37)
            sage: C = M.right_ideals()
            sage: [I.norm() for I in C]
            [16, 32, 32]

            sage: (a,b) = M.quaternion_algebra().invariants()                                       # optional - magma
            sage: magma.eval('A<i,j,k> := QuaternionAlgebra<Rationals() | %s, %s>' % (a,b))         # optional - magma
            ''
            sage: magma.eval('O := QuaternionOrder(%s)' % str(list(C[0].right_order().basis())))    # optional - magma
            ''
            sage: [ magma('rideal<O | %s>' % str(list(I.basis()))).Norm() for I in C]               # optional - magma
            [16, 32, 32]

            sage: A.<i,j,k> = QuaternionAlgebra(-1,-1)
            sage: R = A.ideal([i,j,k,1/2 + 1/2*i + 1/2*j + 1/2*k])      # this is actually an order, so has reduced norm 1
            sage: R.norm()
            1
            sage: [ J.norm() for J in R.cyclic_right_subideals(3) ]     # enumerate maximal right R-ideals of reduced norm 3, verify their norms
            [3, 3, 3, 3]
        """
        G = self.gram_matrix() / QQ(2)
        r = G.det().abs()
        assert r.is_square(), "first is bad!"
        r = r.sqrt()
        # If we know either the left- or the right order, use that one to compute the norm.
        # Otherwise quaternion_order() will raise a RuntimeError and we compute the left order
        try:
            R = self.quaternion_order()
        except RuntimeError:
            R = self.left_order()
        r/= R.discriminant()
        assert r.is_square(), "second is bad!"
        return r.sqrt()

    def conjugate(self):
        """
        Return the ideal with generators the conjugates of the generators for self.

        OUTPUT: a quaternionic fractional ideal

        EXAMPLES::

            sage: I = BrandtModule(3,5).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 34*k, 8*j + 32*k, 40*k)
            sage: I.conjugate()
            Fractional ideal (2 + 2*j + 28*k, 2*i + 4*j + 34*k, 8*j + 32*k, 40*k)
        """
        return self.quaternion_algebra().ideal([b.conjugate() for b in self.basis()],
                                               left_order=self.__right_order,
                                               right_order=self.__left_order)

    def __mul__(self, right):
        """
        Return the product of the fractional ideals ``self`` and ``right``.

        .. note::

           We do not keep track of left or right order structure.

        EXAMPLES::

            sage: I = BrandtModule(3,5).right_ideals()[1]; I
            Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 34*k, 8*j + 32*k, 40*k)
            sage: I*I
            Fractional ideal (8 + 24*j + 16*k, 8*i + 16*j + 136*k, 32*j + 128*k, 160*k)
            sage: I*I.conjugate()
            Fractional ideal (16 + 16*j + 224*k, 8*i + 16*j + 136*k, 32*j + 128*k, 320*k)
            sage: I.multiply_by_conjugate(I)
            Fractional ideal (16 + 16*j + 224*k, 8*i + 16*j + 136*k, 32*j + 128*k, 320*k)
        """
        if not isinstance(right, QuaternionFractionalIdeal_rational):
            return self._scale(right, left=False)
        gens = [a*b for a in self.basis() for b in right.basis()]
        #if self.__right_order == right.__left_order:
        #    left_order = self.__left_order
        #    right_order = right.__right_order
        basis = tuple(basis_for_quaternion_lattice(gens))
        A = self.quaternion_algebra()
        return A.ideal(basis, check=False)

    @cached_method
    def free_module(self):
        r"""
        Return the underlying free `\ZZ`-module corresponding to this ideal.

        OUTPUT:

        Free `\ZZ`-module of rank 4 embedded in an ambient `\QQ^4`.

        EXAMPLES::

            sage: X = BrandtModule(3,5).right_ideals()
            sage: X[0]
            Fractional ideal (2 + 2*j + 8*k, 2*i + 18*k, 4*j + 16*k, 20*k)
            sage: X[0].free_module()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [ 2  0  2  8]
            [ 0  2  0 18]
            [ 0  0  4 16]
            [ 0  0  0 20]
            sage: X[0].scale(1/7).free_module()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [ 2/7    0  2/7  8/7]
            [   0  2/7    0 18/7]
            [   0    0  4/7 16/7]
            [   0    0    0 20/7]

            sage: QuaternionAlgebra(-11,-1).maximal_order().unit_ideal().basis_matrix()
            [ 1/2  1/2    0    0]
            [   0    0  1/2 -1/2]
            [   0    1    0    0]
            [   0    0    0   -1]

        The free module method is also useful since it allows for checking if
        one ideal is contained in another, computing quotients `I/J`, etc.::

            sage: X = BrandtModule(3,17).right_ideals()
            sage: I = X[0].intersection(X[2]); I
            Fractional ideal (2 + 2*j + 164*k, 2*i + 4*j + 46*k, 16*j + 224*k, 272*k)
            sage: I.free_module().is_submodule(X[3].free_module())
            False
            sage: I.free_module().is_submodule(X[1].free_module())
            True
            sage: X[0].free_module() / I.free_module()
            Finitely generated module V/W over Integer Ring with invariants (4, 4)

        This shows that the issue at :trac:`6760` is fixed::

            sage: R.<i,j,k> = QuaternionAlgebra(-1, -13)
            sage: I = R.ideal([2+i, 3*i, 5*j, j+k]); I
            Fractional ideal (2 + i, 3*i, j + k, 5*k)
            sage: I.free_module()
            Free module of degree 4 and rank 4 over Integer Ring
            Echelon basis matrix:
            [2 1 0 0]
            [0 3 0 0]
            [0 0 1 1]
            [0 0 0 5]
        """
        return self.basis_matrix().row_module(ZZ)

    def intersection(self, J):
        """
        Return the intersection of the ideals self and `J`.

        EXAMPLES::

            sage: X = BrandtModule(3,5).right_ideals()
            sage: I = X[0].intersection(X[1]); I
            Fractional ideal (2 + 6*j + 4*k, 2*i + 4*j + 34*k, 8*j + 32*k, 40*k)

        """
        V = self.free_module().intersection(J.free_module())
        H, d = V.basis_matrix()._clear_denom()
        A = self.quaternion_algebra()
        gens = quaternion_algebra_cython.rational_quaternions_from_integral_matrix_and_denom(A, H, d)
        return A.ideal(gens)

    def multiply_by_conjugate(self, J):
        """
        Return product of self and the conjugate Jbar of `J`.

        INPUT:

        - ``J`` -- a quaternion ideal.

        OUTPUT: a quaternionic fractional ideal.

        EXAMPLES::

            sage: R = BrandtModule(3,5).right_ideals()
            sage: R[0].multiply_by_conjugate(R[1])
            Fractional ideal (8 + 8*j + 112*k, 8*i + 16*j + 136*k, 32*j + 128*k, 160*k)
            sage: R[0]*R[1].conjugate()
            Fractional ideal (8 + 8*j + 112*k, 8*i + 16*j + 136*k, 32*j + 128*k, 160*k)
        """
        Jbar = [b.conjugate() for b in J.basis()]
        gens = [a * b for a in self.basis() for b in Jbar]
        basis = tuple(basis_for_quaternion_lattice(gens))
        R = self.quaternion_algebra()
        return R.ideal(basis, check=False)

    def is_equivalent(I, J, B=10):
        """
        Return ``True`` if ``I`` and ``J`` are equivalent as right ideals.

        INPUT:

        - ``I`` -- a fractional quaternion ideal (self)

        - ``J`` -- a fractional quaternion ideal with same order as ``I``

        - ``B`` -- a bound to compute and compare theta series before
          doing the full equivalence test

        OUTPUT: bool

        EXAMPLES::

            sage: R = BrandtModule(3,5).right_ideals(); len(R)
            2
            sage: R[0].is_equivalent(R[1])
            False
            sage: R[0].is_equivalent(R[0])
            True
            sage: OO = R[0].quaternion_order()
            sage: S = OO.right_ideal([3*a for a in R[0].basis()])
            sage: R[0].is_equivalent(S)
            True
        """
        if not isinstance(I, QuaternionFractionalIdeal_rational):
            return False

        if I.right_order() != J.right_order():
            raise ValueError("I and J must be right ideals")

        # Just test theta series first.  If the theta series are
        # different, the ideals are definitely not equivalent.
        if B > 0 and I.theta_series_vector(B) != J.theta_series_vector(B):
            return False

        # The theta series are the same, so perhaps the ideals are
        # equivalent.  We use Prop 1.18 of [Pizer, 1980] to decide.
        # 1. Compute I * Jbar
        # see Prop. 1.17 in Pizer.  Note that we use IJbar instead of
        # JbarI since we work with right ideals
        IJbar = I.multiply_by_conjugate(J)

        # 2. Determine if there is alpha in K such
        #    that N(alpha) = N(I)*N(J) as explained by Pizer.
        c = IJbar.theta_series_vector(2)[1]
        return c != 0

    def __contains__(self, x):
        """
        Return whether x is in self.

        EXAMPLES::

            sage: R.<i,j,k> = QuaternionAlgebra(-3, -13)
            sage: I = R.ideal([2+i, 3*i, 5*j, j+k])
            sage: 2+i in I
            True
            sage: 2+i+j+k in I
            True
            sage: 1+i in I
            False
            sage: 101*j + k in I
            True
        """
        try:
            x = self.quaternion_algebra()(x)
            return self.basis_matrix().transpose().solve_right(vector(x)) in ZZ**4
        except (ValueError, TypeError):
            return False

    @cached_method
    def cyclic_right_subideals(self, p, alpha=None):
        r"""
        Let `I` = ``self``. This function returns the right subideals
        `J` of `I` such that `I/J` is an `\GF{p}`-vector space of
        dimension 2.

        INPUT:

        - ``p`` -- prime number (see below)

        - ``alpha`` -- (default: ``None``) element of quaternion algebra,
          which can be used to parameterize the order of the
          ideals `J`.  More precisely the `J`'s are the right annihilators
          of `(1,0) \alpha^i` for `i=0,1,2,...,p`

        OUTPUT:

        - list of right ideals

        .. NOTE::

           Currently, `p` must satisfy a bunch of conditions, or a
           ``NotImplementedError`` is raised.  In particular, `p` must be
           odd and unramified in the quaternion algebra, must be
           coprime to the index of the right order in the maximal
           order, and also coprime to the normal of self.  (The Brandt
           modules code has a more general algorithm in some cases.)

        EXAMPLES::

            sage: B = BrandtModule(2,37); I = B.right_ideals()[0]
            sage: I.cyclic_right_subideals(3)
            [Fractional ideal (2 + 2*i + 10*j + 90*k, 4*i + 4*j + 152*k, 12*j + 132*k, 444*k), Fractional ideal (2 + 2*i + 2*j + 150*k, 4*i + 8*j + 196*k, 12*j + 132*k, 444*k), Fractional ideal (2 + 2*i + 6*j + 194*k, 4*i + 8*j + 344*k, 12*j + 132*k, 444*k), Fractional ideal (2 + 2*i + 6*j + 46*k, 4*i + 4*j + 4*k, 12*j + 132*k, 444*k)]

            sage: B = BrandtModule(5,389); I = B.right_ideals()[0]
            sage: C = I.cyclic_right_subideals(3); C
            [Fractional ideal (2 + 10*j + 546*k, i + 6*j + 133*k, 12*j + 3456*k, 4668*k), Fractional ideal (2 + 2*j + 2910*k, i + 6*j + 3245*k, 12*j + 3456*k, 4668*k), Fractional ideal (2 + i + 2295*k, 3*i + 2*j + 3571*k, 4*j + 2708*k, 4668*k), Fractional ideal (2 + 2*i + 2*j + 4388*k, 3*i + 2*j + 2015*k, 4*j + 4264*k, 4668*k)]
            sage: [(I.free_module()/J.free_module()).invariants() for J in C]
            [(3, 3), (3, 3), (3, 3), (3, 3)]
            sage: I.scale(3).cyclic_right_subideals(3)
            [Fractional ideal (6 + 30*j + 1638*k, 3*i + 18*j + 399*k, 36*j + 10368*k, 14004*k), Fractional ideal (6 + 6*j + 8730*k, 3*i + 18*j + 9735*k, 36*j + 10368*k, 14004*k), Fractional ideal (6 + 3*i + 6885*k, 9*i + 6*j + 10713*k, 12*j + 8124*k, 14004*k), Fractional ideal (6 + 6*i + 6*j + 13164*k, 9*i + 6*j + 6045*k, 12*j + 12792*k, 14004*k)]
            sage: C = I.scale(1/9).cyclic_right_subideals(3); C
            [Fractional ideal (2/9 + 10/9*j + 182/3*k, 1/9*i + 2/3*j + 133/9*k, 4/3*j + 384*k, 1556/3*k), Fractional ideal (2/9 + 2/9*j + 970/3*k, 1/9*i + 2/3*j + 3245/9*k, 4/3*j + 384*k, 1556/3*k), Fractional ideal (2/9 + 1/9*i + 255*k, 1/3*i + 2/9*j + 3571/9*k, 4/9*j + 2708/9*k, 1556/3*k), Fractional ideal (2/9 + 2/9*i + 2/9*j + 4388/9*k, 1/3*i + 2/9*j + 2015/9*k, 4/9*j + 4264/9*k, 1556/3*k)]
            sage: [(I.scale(1/9).free_module()/J.free_module()).invariants() for J in C]
            [(3, 3), (3, 3), (3, 3), (3, 3)]

            sage: Q.<i,j,k> = QuaternionAlgebra(-2,-5)
            sage: I = Q.ideal([Q(1),i,j,k])
            sage: I.cyclic_right_subideals(3)
            [Fractional ideal (1 + 2*j, i + k, 3*j, 3*k), Fractional ideal (1 + j, i + 2*k, 3*j, 3*k), Fractional ideal (1 + 2*i, 3*i, j + 2*k, 3*k), Fractional ideal (1 + i, 3*i, j + k, 3*k)]

        The general algorithm is not yet implemented here::

            sage: I.cyclic_right_subideals(3)[0].cyclic_right_subideals(3)
            Traceback (most recent call last):
            ...
            NotImplementedError: general algorithm not implemented (The given basis vectors must be linearly independent.)
        """
        R = self.right_order()
        Q = self.quaternion_algebra()
        f = Q.modp_splitting_map(p)
        if alpha is not None:
            alpha = f(alpha)
        W = GF(p)**4
        try:
            A = W.span_of_basis([W(f(a).list()) for a in self.basis()])
            scale = 1
            IB = self.basis_matrix()
        except (ValueError, ZeroDivisionError):
            # try rescaling the ideal.
            B, d = self.basis_matrix()._clear_denom()
            g = gcd(B.list())
            IB = B / g
            scale = g / d
            try:
                A = W.span_of_basis([W(f(Q(a.list())).list()) for a in IB.rows()])
            except (ValueError, ZeroDivisionError) as msg:
                # Here we could replace the ideal by an *equivalent*
                # ideal that works.  This is always possible.
                # However, I haven't implemented that algorithm yet.
                raise NotImplementedError("general algorithm not implemented (%s)" % msg)

        Ai = A.basis_matrix()**(-1)
        AiB = Ai.change_ring(QQ) * IB

        # Do not care about the denominator since we're really working in I/p*I.
        AiB, _ = AiB._clear_denom()

        pB = p*IB
        pB, d = pB._clear_denom()

        ans = []
        Z = matrix(ZZ,2,4)

        P1 = P1List(p)
        if alpha is None:
            lines = P1
        else:
            x = alpha
            lines = []
            for i in range(p+1):
                lines.append(P1.normalize(x[0,0], x[0,1]))
                x *= alpha

        for u,v in lines:
            # The following does:
            #    z = matrix(QQ,2,4,[0,-v,0,u, -v,0,u,0],check=False) * AiB
            Z[0,1]=-v; Z[0,3]=u; Z[1,0]=-v; Z[1,2]=u
            z = Z * AiB
            # Now construct submodule of the ideal I spanned by the
            # linear combinations given by z of the basis for J along
            # with p*I.
            G = (d*z).stack(pB)  # have to multiply by d since we divide by it below in the "gens = " line.
            H = G._hnf_pari(0, include_zero_rows=False)
            gens = tuple(quaternion_algebra_cython.rational_quaternions_from_integral_matrix_and_denom(Q, H, d))
            if scale != 1:
                gens = tuple([scale * gg for gg in gens])
            J = R.right_ideal(gens, check=False)
            ans.append(J)
        return ans

#######################################################################
# Some utility functions that are needed here and are too
# specialized to go elsewhere.
#######################################################################

def basis_for_quaternion_lattice(gens, reverse = False):
    r"""
    Return a basis for the `\ZZ`-lattice in a quaternion algebra
    spanned by the given gens.

    INPUT:

    - ``gens`` -- list of elements of a single quaternion algebra

    - ``reverse`` -- when computing the HNF do it on the basis
      `(k,j,i,1)` instead of `(1,i,j,k)`; this ensures
      that if ``gens`` are the generators for an order,
      the first returned basis vector is 1

    EXAMPLES::

        sage: from sage.algebras.quatalg.quaternion_algebra import basis_for_quaternion_lattice
        sage: A.<i,j,k> = QuaternionAlgebra(-1,-7)
        sage: basis_for_quaternion_lattice([i+j, i-j, 2*k, A(1/3)])
        [1/3, i + j, 2*j, 2*k]

        sage: basis_for_quaternion_lattice([A(1),i,j,k])
        [1, i, j, k]

    """
    if not gens:
        return []
    Z, d = quaternion_algebra_cython.integral_matrix_and_denom_from_rational_quaternions(gens, reverse)
    H = Z._hnf_pari(0, include_zero_rows=False)
    A = gens[0].parent()
    return quaternion_algebra_cython.rational_quaternions_from_integral_matrix_and_denom(A, H, d, reverse)


def intersection_of_row_modules_over_ZZ(v):
    r"""
    Intersect the `\ZZ`-modules with basis matrices the full rank `4 \times 4`
    `\QQ`-matrices in the list v.

    The returned intersection is
    represented by a `4 \times 4` matrix over `\QQ`.  This can also be done
    using modules and intersection, but that would take over twice as long
    because of overhead, hence this function.

    EXAMPLES::

        sage: a = matrix(QQ,4,[-2, 0, 0, 0, 0, -1, -1, 1, 2, -1/2, 0, 0, 1, 1, -1, 0])
        sage: b = matrix(QQ,4,[0, -1/2, 0, -1/2, 2, 1/2, -1, -1/2, 1, 2, 1, -2, 0, -1/2, -2, 0])
        sage: c = matrix(QQ,4,[0, 1, 0, -1/2, 0, 0, 2, 2, 0, -1/2, 1/2, -1, 1, -1, -1/2, 0])
        sage: v = [a,b,c]
        sage: from sage.algebras.quatalg.quaternion_algebra import intersection_of_row_modules_over_ZZ
        sage: M = intersection_of_row_modules_over_ZZ(v); M
        [   2    0   -1   -1]
        [  -4    1    1   -3]
        [  -3 19/2   -1   -4]
        [   2   -3   -8    4]
        sage: M2 = a.row_module(ZZ).intersection(b.row_module(ZZ)).intersection(c.row_module(ZZ))
        sage: M.row_module(ZZ) == M2
        True
    """
    if len(v) <= 0:
        raise ValueError("v must have positive length")
    if len(v) == 1:
        return v[0]
    elif len(v) == 2:
        # real work - the base case
        a, b = v
        s,_ = a.stack(b)._clear_denom()
        s = s.transpose()
        K = s.right_kernel_matrix(algorithm='pari', basis='computed')
        n = a.nrows()
        return K.matrix_from_columns(range(n)) * a
    else:
        # induct
        w = intersection_of_row_modules_over_ZZ(v[:2])
        return intersection_of_row_modules_over_ZZ([w] + v[2:])


def normalize_basis_at_p(e, p, B = lambda x,y: (x*y.conjugate()).reduced_trace()):
    r"""
    Compute a (at ``p``) normalized basis from the given basis ``e``
    of a `\ZZ`-module.

    The returned basis is (at ``p``) a `\ZZ_p` basis for the same
    module, and has the property that with respect to it the quadratic
    form induced by the bilinear form B is represented as a orthogonal
    sum of atomic forms multiplied by p-powers.

    If `p \neq 2` this means that the form is diagonal with respect to
    this basis.

    If `p = 2` there may be additional 2-dimensional subspaces on which
    the form is represented as `2^e (ax^2 + bxy + cx^2)` with
    `0 = v_2(b) = v_2(a) \leq v_2(c)`.

    INPUT:

    - ``e`` -- list; basis of a `\ZZ` module.
      WARNING: will be modified!

    - ``p`` -- prime for at which the basis should be normalized

    - ``B`` -- (default:
      ``lambda x,y: ((x*y).conjugate()).reduced_trace()``)
      a bilinear form with respect to which to normalize

    OUTPUT:

    - A list containing two-element tuples: The first element of
      each tuple is a basis element, the second the valuation of
      the orthogonal summand to which it belongs. The list is sorted
      by ascending valuation.

    EXAMPLES::

        sage: from sage.algebras.quatalg.quaternion_algebra import normalize_basis_at_p
        sage: A.<i,j,k> = QuaternionAlgebra(-1, -1)
        sage: e = [A(1), i, j, k]
        sage: normalize_basis_at_p(e, 2)
        [(1, 0), (i, 0), (j, 0), (k, 0)]

        sage: A.<i,j,k> = QuaternionAlgebra(210)
        sage: e = [A(1), i, j, k]
        sage: normalize_basis_at_p(e, 2)
        [(1, 0), (i, 1), (j, 1), (k, 2)]

        sage: A.<i,j,k> = QuaternionAlgebra(286)
        sage: e = [A(1), k, 1/2*j + 1/2*k, 1/2 + 1/2*i + 1/2*k]
        sage: normalize_basis_at_p(e, 5)
        [(1, 0), (1/2*j + 1/2*k, 0), (-5/6*j + 1/6*k, 1), (1/2*i, 1)]

        sage: A.<i,j,k> = QuaternionAlgebra(-1,-7)
        sage: e = [A(1), k, j, 1/2 + 1/2*i + 1/2*j + 1/2*k]
        sage: normalize_basis_at_p(e, 2)
        [(1, 0), (1/2 + 1/2*i + 1/2*j + 1/2*k, 0), (-34/105*i - 463/735*j + 71/105*k, 1), (-34/105*i - 463/735*j + 71/105*k, 1)]
    """

    N = len(e)
    if N == 0:
        return []
    else:
        min_m, min_n, min_v = 0, 0, infinity

        # Find two basis vector on which the bilinear form has minimal
        # p-valuation. If there is more than one such pair, always
        # prefer diagonal entries over any other and (secondary) take
        # min_m and then min_n as small as possible
        for m in range(N):
            for n in range(m, N):
                v = B(e[m], e[n]).valuation(p)
                if v < min_v or (v == min_v and (min_m != min_n) and (m == n)):
                    min_m, min_n, min_v = m, n, v


        if (min_m == min_n) or p != 2:      # In this case we can diagonalize
            if min_m == min_n:              # Diagonal entry has minimal valuation
                f0 = e[min_m]
            else:
                f0 = e[min_m] + e[min_n]    # Only off-diagonal entries have min. val., but p!=2

            # Swap with first vector
            e[0], e[min_m] = e[min_m], e[0]

            # Orthogonalize remaining vectors with respect to f
            c = B(f0, f0)
            for l in range(1, N):
                e[l] = e[l] - B(e[l],f0)/c * f0

            # Recursively normalize remaining vectors
            f = normalize_basis_at_p(e[1:], p)
            f.insert(0, (f0, min_v - valuation(p, 2)))
            return f

        else:   # p = 2 and only off-diagonal entries have min. val., gives 2-dim. block
            # first diagonal entry should have smaller valuation
            if B(e[min_m],e[min_m]).valuation(p) > B(e[min_n],e[min_n]).valuation(p):
                e[min_m], e[min_n] = e[min_n], e[min_m]

            f0 = p**min_v / B(e[min_m],e[min_n]) * e[min_m]
            f1 = e[min_n]

            # Ensures that (B(f0,f0)/2).valuation(p) <= B(f0,f1).valuation(p)
            if B(f0,f1).valuation(p) + 1 < B(f0,f0).valuation(p):
                f0 = f0 + f1
                f1 = f0

            # Make remaining vectors orthogonal to span of f0, f1
            e[min_m] = e[0]
            e[min_n] = e[1]

            B00 = B(f0,f0)
            B11 = B(f1,f1)
            B01 = B(f0,f1)
            d = B00*B11 - B01**2
            tu = [ (B01 * B(f1,e[l]) - B11 * B(f0,e[l]),
                    B01 * B(f0,e[l]) - B00 * B(f1,e[l])) for l in range(2,N) ]

            e[2:n] = [ e[l] + tu[l-2][0]/d * f0 + tu[l-2][1]/d * f1 for l in range(2,N) ]

            # Recursively normalize remaining vectors
            f = normalize_basis_at_p(e[2:N], p)
            return [(f0, min_v), (f1, min_v)] + f


def maxord_solve_aux_eq(a, b, p):
    r"""
    Given ``a`` and ``b`` and an even prime ideal ``p`` find
    (y,z,w) with y a unit mod `p^{2e}` such that

    .. MATH::

        1 - ay^2 - bz^2 + abw^2 \equiv 0 mod p^{2e},

    where `e` is the ramification index of `p`.

    Currently only `p=2` is implemented by hardcoding solutions.

    INPUT:

    - ``a`` -- integer with `v_p(a) = 0`

    - ``b`` -- integer with `v_p(b) \in \{0,1\}`

    - ``p`` -- even prime ideal (actually only ``p=ZZ(2)`` is implemented)

    OUTPUT:

    - A tuple `(y, z, w)`

    EXAMPLES::

        sage: from sage.algebras.quatalg.quaternion_algebra import maxord_solve_aux_eq
        sage: for a in [1,3]:
        ....:     for b in [1,2,3]:
        ....:         (y,z,w) = maxord_solve_aux_eq(a, b, 2)
        ....:         assert mod(y, 4) == 1 or mod(y, 4) == 3
        ....:         assert mod(1 - a*y^2 - b*z^2 + a*b*w^2, 4) == 0
    """
    if p != ZZ(2):
        raise NotImplementedError("Algorithm only implemented over ZZ at the moment")

    v_a = a.valuation(p)
    v_b = b.valuation(p)

    if v_a != 0:
        raise RuntimeError("a must have v_p(a)=0")
    if v_b != 0 and v_b != 1:
        raise RuntimeError("b must have v_p(b) in {0,1}")

    R = ZZ.quo(ZZ(4))
    lut = {
        (R(1), R(1)) : (1,1,1),
        (R(1), R(2)) : (1,0,0),
        (R(1), R(3)) : (1,0,0),
        (R(3), R(1)) : (1,1,1),
        (R(3), R(2)) : (1,0,1),
        (R(3), R(3)) : (1,1,1), }

    return lut[(R(a), R(b))]