Quaternion Algebras¶

AUTHORS:

Jon Bobber – 2009 rewrite
William Stein – 2009 rewrite

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

sage.algebras.quatalg.quaternion_algebra.QuaternionAlgebra(arg0, arg1=None, arg2=None, names='i, j, k')¶

There are three input formats:

QuaternionAlgebra(a, b): quaternion algebra generated by i, j subject to $i^2 = a$ , $j^2 = b$ , $j * i = -i * j$ .
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 $k = i * j$ .
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 $j*i=-i*j$ :

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
sage: QuaternionAlgebra(sqrt(-1), sqrt(-3))
Quaternion Algebra (I, sqrt(-3)) with base ring Symbolic Ring
sage: QuaternionAlgebra(0,0)
...
ValueError: a and b must be nonzero
sage: QuaternionAlgebra(GF(2)(1),1)
...
ValueError: a and b must be elements of a field 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 $i*j=-j*i$ :

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

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

class sage.algebras.quatalg.quaternion_algebra.QuaternionAlgebra_ab(base_ring, a, b, names='i, j, k')¶

Bases: sage.algebras.quatalg.quaternion_algebra.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.

EXAMPLES:

sage: QuaternionAlgebra(QQ, -7, -21)  # indirect doctest
Quaternion Algebra (-7, -21) with base ring Rational Field

discriminant()¶

Given a quaternion algebra $A$ defined over the field of rational numbers, 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

This raises a NotImplementedError if the base field is not the rational numbers:

sage: QuaternionAlgebra(QQ[sqrt(2)],3,19).discriminant()
...
NotImplementedError: base field must be rational numbers

gen(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]

ideal(gens, left_order=None, right_order=None, check=True)¶

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)

inner_product_matrix()¶

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]

invariants()¶

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

maximal_order()¶

Return a maximal order in this quaternion algebra.

OUTPUT: an 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)

modp_splitting_data(*args, **kwds)¶

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)
...
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)
...
NotImplementedError: p must be odd            

modp_splitting_map(p)¶

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]

quaternion_order(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]

ramified_primes()¶

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]

class sage.algebras.quatalg.quaternion_algebra.QuaternionAlgebra_abstract¶

Bases: sage.rings.ring.Algebra

basis()¶

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

inner_product_matrix()¶

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 $2$ , $2a$ , $2b$ , $2ab$ .

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]

is_commutative()¶

Return False always, since all quaternion algebras are noncommutative.

EXAMPLES:

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

is_division_algebra()¶

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()
...
NotImplementedError: base field must be rational numbers

is_exact()¶

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

is_field(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

is_finite()¶

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

is_integral_domain(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

is_matrix_ring()¶

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()
...
NotImplementedError: base field must be rational numbers

is_noetherian()¶

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

ngens()¶

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]

order()¶

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

random_element(*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: QuaternionAlgebra(QQ[sqrt(2)],-3,7).random_element()
(sqrt2 + 2)*i + (-12*sqrt2 - 2)*j + (-sqrt2 + 1)*k
sage: QuaternionAlgebra(-3,19).random_element()
-1 + 2*i - j - 6/5*k
sage: QuaternionAlgebra(GF(17)(2),3).random_element()
14 + 10*i + 4*j + 7*k

Specify the numerator and denominator bounds:

sage: QuaternionAlgebra(-3,19).random_element(10^6,10^6)
-979933/553629 + 255525/657688*i - 3511/6929*j - 700105/258683*k

vector_space()¶

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]

class sage.algebras.quatalg.quaternion_algebra.QuaternionFractionalIdeal(ring, gens, coerce=True)¶: Bases: sage.rings.ideal.Ideal_fractional

class sage.algebras.quatalg.quaternion_algebra.QuaternionFractionalIdeal_rational(basis, left_order=None, right_order=None, check=True)¶

Bases: sage.algebras.quatalg.quaternion_algebra.QuaternionFractionalIdeal

A fractional ideal in a rational quaternion algebra.

basis()¶

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*j, 1/2*i + 1/2*k, j, k)

basis_matrix()¶

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   0 1/2   0]
[  0 1/2   0 1/2]
[  0   0   1   0]
[  0   0   0   1]

conjugate()¶

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)

cyclic_right_subideals(*args, **kwds)¶

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)
...
NotImplementedError: general algorithm not implemented (The given basis vectors must be linearly independent.)

free_module(*args, **kwds)¶

Return the underlying free $\ZZ$ -module corresponding to this ideal.

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]

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)

gens()¶

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*j, 1/2*i + 1/2*k, j, k)

gram_matrix()¶

Return the Gram matrix of this fractional ideal.

OUTPUT: 4x4 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]

intersection(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)

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

left_order()¶

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: print [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]

multiply_by_conjugate(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)

norm()¶

Return the norm of this fractional ideal.

OUTPUT: rational number

EXAMPLES:

sage: C = BrandtModule(37).right_ideals()
sage: [I.norm() for I in C]
[32, 64, 64]

quadratic_form()¶

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)

quaternion_algebra()¶

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

quaternion_order()¶

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

right_order()¶

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

ring()¶

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

scale(alpha, left=False)¶

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

theta_series(B, var='q')¶

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

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

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()
64
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)

theta_series_vector(B)¶

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)

class sage.algebras.quatalg.quaternion_algebra.QuaternionOrder(A, basis, check=True)¶

Bases: sage.rings.ring.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'>

basis()¶

Return fix choice of basis for this quaternion order.

EXAMPLES:

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

discriminant()¶

Return the discriminant of this order, which we define 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())
<type 'sage.rings.rational.Rational'>

free_module()¶

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*j, 1/2*i + 1/2*k, j, k)
sage: R.free_module()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/2   0 1/2   0]
[  0 1/2   0 1/2]
[  0   0   1   0]
[  0   0   0   1]

gen(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*j, 1/2*i + 1/2*k, j, k)
sage: R.gen(0)
1/2 + 1/2*j
sage: R.gen(1)
1/2*i + 1/2*k
sage: R.gen(2)
j
sage: R.gen(3)
k

gens()¶

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)

intersection(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*j, 1/2*i + 1/2*k, j, 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)

left_ideal(gens, check=True)¶

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 + j, i + k, 2*j, 2*k)

ngens()¶

Return the number of generators (which is 4).

EXAMPLES:

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

quadratic_form()¶

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)

quaternion_algebra()¶

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

random_element(*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

$ae_1 + be_2 + ce_3 + de_4$

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()
-4 + i - 4*j + k
sage: QuaternionAlgebra(-11,-1).maximal_order().random_element(-10,10)
-9/2 - 7/2*i - 7/2*j + 3/2*k

right_ideal(gens, check=True)¶

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 + j, i + k, 2*j, 2*k)

ternary_quadratic_form(include_basis=False)¶

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)

unit_ideal()¶

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*j, 1/2*i + 1/2*k, j, k)

sage.algebras.quatalg.quaternion_algebra.basis_for_quaternion_lattice(gens)¶

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

EXAMPLES:

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

sage.algebras.quatalg.quaternion_algebra.intersection_of_row_modules_over_ZZ(v)¶

Intersects the $\ZZ$ -modules with basis matrices the full rank 4x4 $\QQ$ -matrices in the list v. The returned intersection is represented by a 4x4 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

sage.algebras.quatalg.quaternion_algebra.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

sage.algebras.quatalg.quaternion_algebra.unpickle_QuaternionAlgebra_v0(*key)¶

The 0th version of pickling for quaternion algebras.

EXAMPLES:

sage: Q = QuaternionAlgebra(-5,-19)
sage: f, t = Q.__reduce__()
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

Quaternion Algebras¶

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