Elements of Quaternion Algebras¶

Sage allows for computation with elements of quaternion algebras over a nearly arbitrary base field of characteristic not 2. Sage also has very highly optimized implementation of arithmetic in rational quaternion algebras and quaternion algebras over number fields.

class sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_abstract[source]¶

Bases: AlgebraElement

coefficient_tuple()[source]¶

Return 4-tuple of coefficients of this quaternion.

EXAMPLES:

Sage

sage: K.<x> = QQ['x']
sage: Q.<i,j,k> = QuaternionAlgebra(Frac(K),-5,-2)
sage: a = 1/2*x^2 + 2/3*x*i - 3/4*j + 5/7*k
sage: type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic'>
sage: a.coefficient_tuple()
(1/2*x^2, 2/3*x, -3/4, 5/7)

Python

>>> from sage.all import *
>>> K = QQ['x']; (x,) = K._first_ngens(1)
>>> Q = QuaternionAlgebra(Frac(K),-Integer(5),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)
>>> a = Integer(1)/Integer(2)*x**Integer(2) + Integer(2)/Integer(3)*x*i - Integer(3)/Integer(4)*j + Integer(5)/Integer(7)*k
>>> type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic'>
>>> a.coefficient_tuple()
(1/2*x^2, 2/3*x, -3/4, 5/7)

conjugate()[source]¶

Return the conjugate of the quaternion: if \(\theta = x + yi + zj + wk\), return \(x - yi - zj - wk\); that is, return theta.reduced_trace() - theta.

EXAMPLES:

Sage

sage: A.<i,j,k> = QuaternionAlgebra(QQ,-5,-2)
sage: a = 3*i - j + 2
sage: type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field'>
sage: a.conjugate()
2 - 3*i + j

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(QQ,-Integer(5),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> a = Integer(3)*i - j + Integer(2)
>>> type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field'>
>>> a.conjugate()
2 - 3*i + j

The “universal” test:

Sage

sage: K.<x,y,z,w,a,b> = QQ[]
sage: Q.<i,j,k> = QuaternionAlgebra(a,b)
sage: theta = x+y*i+z*j+w*k
sage: theta.conjugate()
x + (-y)*i + (-z)*j + (-w)*k

Python

>>> from sage.all import *
>>> K = QQ['x, y, z, w, a, b']; (x, y, z, w, a, b,) = K._first_ngens(6)
>>> Q = QuaternionAlgebra(a,b, names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)
>>> theta = x+y*i+z*j+w*k
>>> theta.conjugate()
x + (-y)*i + (-z)*j + (-w)*k

is_constant()[source]¶

Return True if this quaternion is constant, i.e., has no \(i\), \(j\), or \(k\) term.

OUTPUT: boolean

EXAMPLES:

Sage

sage: A.<i,j,k> = QuaternionAlgebra(-1,-2)
sage: A(1).is_constant()
True
sage: A(1+i).is_constant()
False
sage: A(i).is_constant()
False

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> A(Integer(1)).is_constant()
True
>>> A(Integer(1)+i).is_constant()
False
>>> A(i).is_constant()
False

matrix(action='right')[source]¶

Return the matrix of right or left multiplication of self on the basis for the ambient quaternion algebra.

In particular, if action is 'right' (the default), returns the matrix of the mapping sending x to x*self.

INPUT:

action – (default: 'right') 'right' or 'left'

OUTPUT: a matrix

EXAMPLES:

Sage

sage: Q.<i,j,k> = QuaternionAlgebra(-3,-19)
sage: a = 2/3 -1/2*i + 3/5*j - 4/3*k
sage: a.matrix()
[  2/3  -1/2   3/5  -4/3]
[  3/2   2/3     4   3/5]
[-57/5 -76/3   2/3   1/2]
[   76 -57/5  -3/2   2/3]
sage: a.matrix() == a.matrix(action='right')
True
sage: a.matrix(action='left')
[  2/3  -1/2   3/5  -4/3]
[  3/2   2/3    -4  -3/5]
[-57/5  76/3   2/3  -1/2]
[   76  57/5   3/2   2/3]
sage: (i*a,j*a,k*a)
(3/2 + 2/3*i + 4*j + 3/5*k, -57/5 - 76/3*i + 2/3*j + 1/2*k, 76 - 57/5*i - 3/2*j + 2/3*k)
sage: a.matrix(action='foo')
Traceback (most recent call last):
...
ValueError: action must be either 'left' or 'right'

Python

>>> from sage.all import *
>>> Q = QuaternionAlgebra(-Integer(3),-Integer(19), names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)
>>> a = Integer(2)/Integer(3) -Integer(1)/Integer(2)*i + Integer(3)/Integer(5)*j - Integer(4)/Integer(3)*k
>>> a.matrix()
[  2/3  -1/2   3/5  -4/3]
[  3/2   2/3     4   3/5]
[-57/5 -76/3   2/3   1/2]
[   76 -57/5  -3/2   2/3]
>>> a.matrix() == a.matrix(action='right')
True
>>> a.matrix(action='left')
[  2/3  -1/2   3/5  -4/3]
[  3/2   2/3    -4  -3/5]
[-57/5  76/3   2/3  -1/2]
[   76  57/5   3/2   2/3]
>>> (i*a,j*a,k*a)
(3/2 + 2/3*i + 4*j + 3/5*k, -57/5 - 76/3*i + 2/3*j + 1/2*k, 76 - 57/5*i - 3/2*j + 2/3*k)
>>> a.matrix(action='foo')
Traceback (most recent call last):
...
ValueError: action must be either 'left' or 'right'

We test over a more generic base field:

Sage

sage: K.<x> = QQ['x']
sage: Q.<i,j,k> = QuaternionAlgebra(Frac(K),-5,-2)
sage: a = 1/2*x^2 + 2/3*x*i - 3/4*j + 5/7*k
sage: type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic'>
sage: a.matrix()
[1/2*x^2   2/3*x    -3/4     5/7]
[-10/3*x 1/2*x^2   -25/7    -3/4]
[    3/2    10/7 1/2*x^2  -2/3*x]
[  -50/7     3/2  10/3*x 1/2*x^2]

Python

>>> from sage.all import *
>>> K = QQ['x']; (x,) = K._first_ngens(1)
>>> Q = QuaternionAlgebra(Frac(K),-Integer(5),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)
>>> a = Integer(1)/Integer(2)*x**Integer(2) + Integer(2)/Integer(3)*x*i - Integer(3)/Integer(4)*j + Integer(5)/Integer(7)*k
>>> type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic'>
>>> a.matrix()
[1/2*x^2   2/3*x    -3/4     5/7]
[-10/3*x 1/2*x^2   -25/7    -3/4]
[    3/2    10/7 1/2*x^2  -2/3*x]
[  -50/7     3/2  10/3*x 1/2*x^2]

pair(right)[source]¶

Return the result of pairing self and right, which should both be elements of a quaternion algebra. The pairing is (x,y) = (x.conjugate()*y).reduced_trace().

INPUT:

right – quaternion

EXAMPLES:

Sage

sage: A.<i,j,k>=QuaternionAlgebra(-1,-2)
sage: (1+i+j-2*k).pair(2/3+5*i-3*j+k)
-26/3
sage: x = 1+i+j-2*k; y = 2/3+5*i-3*j+k
sage: x.pair(y)
-26/3
sage: y.pair(x)
-26/3
sage: (x.conjugate()*y).reduced_trace()
-26/3

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> (Integer(1)+i+j-Integer(2)*k).pair(Integer(2)/Integer(3)+Integer(5)*i-Integer(3)*j+k)
-26/3
>>> x = Integer(1)+i+j-Integer(2)*k; y = Integer(2)/Integer(3)+Integer(5)*i-Integer(3)*j+k
>>> x.pair(y)
-26/3
>>> y.pair(x)
-26/3
>>> (x.conjugate()*y).reduced_trace()
-26/3

reduced_characteristic_polynomial(var='x')[source]¶

Return the reduced characteristic polynomial of this quaternion algebra element, which is \(X^2 - tX + n\), where \(t\) is the reduced trace and \(n\) is the reduced norm.

INPUT:

var – string (default: 'x'); indeterminate of characteristic polynomial

EXAMPLES:

Sage

sage: A.<i,j,k>=QuaternionAlgebra(-1,-2)
sage: i.reduced_characteristic_polynomial()
x^2 + 1
sage: j.reduced_characteristic_polynomial()
x^2 + 2
sage: (i+j).reduced_characteristic_polynomial()
x^2 + 3
sage: (2+j+k).reduced_trace()
4
sage: (2+j+k).reduced_characteristic_polynomial('T')
T^2 - 4*T + 8

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> i.reduced_characteristic_polynomial()
x^2 + 1
>>> j.reduced_characteristic_polynomial()
x^2 + 2
>>> (i+j).reduced_characteristic_polynomial()
x^2 + 3
>>> (Integer(2)+j+k).reduced_trace()
4
>>> (Integer(2)+j+k).reduced_characteristic_polynomial('T')
T^2 - 4*T + 8

reduced_norm()[source]¶

Return the reduced norm of self: if \(\theta = x + yi + zj + wk\), then \(\theta\) has reduced norm \(x^2 - ay^2 - bz^2 + abw^2\).

EXAMPLES:

Sage

sage: K.<x,y,z,w,a,b> = QQ[]
sage: Q.<i,j,k> = QuaternionAlgebra(a,b)
sage: theta = x+y*i+z*j+w*k
sage: theta.reduced_norm()
w^2*a*b - y^2*a - z^2*b + x^2

Python

>>> from sage.all import *
>>> K = QQ['x, y, z, w, a, b']; (x, y, z, w, a, b,) = K._first_ngens(6)
>>> Q = QuaternionAlgebra(a,b, names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)
>>> theta = x+y*i+z*j+w*k
>>> theta.reduced_norm()
w^2*a*b - y^2*a - z^2*b + x^2

reduced_trace()[source]¶

Return the reduced trace of self: if \(\theta = x + yi + zj + wk\), then \(\theta\) has reduced trace \(2x\).

EXAMPLES:

Sage

sage: K.<x,y,z,w,a,b> = QQ[]
sage: Q.<i,j,k> = QuaternionAlgebra(a,b)
sage: theta = x+y*i+z*j+w*k
sage: theta.reduced_trace()
2*x

Python

>>> from sage.all import *
>>> K = QQ['x, y, z, w, a, b']; (x, y, z, w, a, b,) = K._first_ngens(6)
>>> Q = QuaternionAlgebra(a,b, names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)
>>> theta = x+y*i+z*j+w*k
>>> theta.reduced_trace()
2*x

class sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic[source]¶: Bases: QuaternionAlgebraElement_abstract

class sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_number_field[source]¶

Bases: QuaternionAlgebraElement_abstract

EXAMPLES:

Sage

sage: K.<a> = QQ[2^(1/3)]; Q.<i,j,k> = QuaternionAlgebra(K,-a,a+1)          # needs sage.symbolic
sage: Q([a,-2/3,a^2-1/2,a*2])           # implicit doctest                  # needs sage.symbolic
a + (-2/3)*i + (a^2 - 1/2)*j + 2*a*k

Python

>>> from sage.all import *
>>> K = QQ[Integer(2)**(Integer(1)/Integer(3))]; (a,) = K._first_ngens(1); Q = QuaternionAlgebra(K,-a,a+Integer(1), names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3)# needs sage.symbolic
>>> Q([a,-Integer(2)/Integer(3),a**Integer(2)-Integer(1)/Integer(2),a*Integer(2)])           # implicit doctest                  # needs sage.symbolic
a + (-2/3)*i + (a^2 - 1/2)*j + 2*a*k

class sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field[source]¶

Bases: QuaternionAlgebraElement_abstract

coefficient_tuple()[source]¶

Return 4-tuple of rational numbers which are the coefficients of this quaternion.

EXAMPLES:

Sage

sage: A.<i,j,k> = QuaternionAlgebra(-1,-2)
sage: (2/3 + 3/5*i + 4/3*j - 5/7*k).coefficient_tuple()
(2/3, 3/5, 4/3, -5/7)

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> (Integer(2)/Integer(3) + Integer(3)/Integer(5)*i + Integer(4)/Integer(3)*j - Integer(5)/Integer(7)*k).coefficient_tuple()
(2/3, 3/5, 4/3, -5/7)

conjugate()[source]¶

Return the conjugate of this quaternion.

EXAMPLES:

Sage

sage: A.<i,j,k> = QuaternionAlgebra(QQ,-5,-2)
sage: a = 3*i - j + 2
sage: type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field'>
sage: a.conjugate()
2 - 3*i + j
sage: b = 1 + 1/3*i + 1/5*j - 1/7*k
sage: b.conjugate()
1 - 1/3*i - 1/5*j + 1/7*k

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(QQ,-Integer(5),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> a = Integer(3)*i - j + Integer(2)
>>> type(a)
<class 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_rational_field'>
>>> a.conjugate()
2 - 3*i + j
>>> b = Integer(1) + Integer(1)/Integer(3)*i + Integer(1)/Integer(5)*j - Integer(1)/Integer(7)*k
>>> b.conjugate()
1 - 1/3*i - 1/5*j + 1/7*k

denominator()[source]¶

Return the lowest common multiple of the denominators of the coefficients of i, j and k for this quaternion.

EXAMPLES:

Sage

sage: A = QuaternionAlgebra(QQ, -1, -1)
sage: A.<i,j,k> = QuaternionAlgebra(QQ, -1, -1)
sage: a = (1/2) + (1/5)*i + (5/12)*j + (1/13)*k
sage: a
1/2 + 1/5*i + 5/12*j + 1/13*k
sage: a.denominator()
780
sage: lcm([2, 5, 12, 13])
780
sage: (a * a).denominator()
608400
sage: (a + a).denominator()
390

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(QQ, -Integer(1), -Integer(1))
>>> A = QuaternionAlgebra(QQ, -Integer(1), -Integer(1), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> a = (Integer(1)/Integer(2)) + (Integer(1)/Integer(5))*i + (Integer(5)/Integer(12))*j + (Integer(1)/Integer(13))*k
>>> a
1/2 + 1/5*i + 5/12*j + 1/13*k
>>> a.denominator()
780
>>> lcm([Integer(2), Integer(5), Integer(12), Integer(13)])
780
>>> (a * a).denominator()
608400
>>> (a + a).denominator()
390

denominator_and_integer_coefficient_tuple()[source]¶

Return 5-tuple d, x, y, z, w, where this rational quaternion is equal to \((x + yi + zj + wk)/d\) and x, y, z, w do not share a common factor with d.

OUTPUT: 5-tuple of Integers

EXAMPLES:

Sage

sage: A.<i,j,k>=QuaternionAlgebra(-1,-2)
sage: (2 + 3*i + 4/3*j - 5*k).denominator_and_integer_coefficient_tuple()
(3, 6, 9, 4, -15)

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> (Integer(2) + Integer(3)*i + Integer(4)/Integer(3)*j - Integer(5)*k).denominator_and_integer_coefficient_tuple()
(3, 6, 9, 4, -15)

integer_coefficient_tuple()[source]¶

Return the integer part of this quaternion, ignoring the common denominator.

OUTPUT: 4-tuple of Integers

EXAMPLES:

Sage

sage: A.<i,j,k>=QuaternionAlgebra(-1,-2)
sage: (2 + 3*i + 4/3*j - 5*k).integer_coefficient_tuple()
(6, 9, 4, -15)

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> (Integer(2) + Integer(3)*i + Integer(4)/Integer(3)*j - Integer(5)*k).integer_coefficient_tuple()
(6, 9, 4, -15)

is_constant()[source]¶

Return True if this quaternion is constant, i.e., has no \(i\), \(j\), or \(k\) term.

OUTPUT: boolean

EXAMPLES:

Sage

sage: A.<i,j,k>=QuaternionAlgebra(-1,-2)
sage: A(1/3).is_constant()
True
sage: A(-1).is_constant()
True
sage: (1+i).is_constant()
False
sage: j.is_constant()
False

Python

>>> from sage.all import *
>>> A = QuaternionAlgebra(-Integer(1),-Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = A._first_ngens(3)
>>> A(Integer(1)/Integer(3)).is_constant()
True
>>> A(-Integer(1)).is_constant()
True
>>> (Integer(1)+i).is_constant()
False
>>> j.is_constant()
False

reduced_norm()[source]¶

Return the reduced norm of self.

Given a quaternion \(x+yi+zj+wk\), this is \(x^2 - ay^2 - bz^2 + abw^2\).

EXAMPLES:

Sage

sage: K.<i,j,k> = QuaternionAlgebra(QQ, -5, -2)
sage: i.reduced_norm()
5
sage: j.reduced_norm()
2
sage: a = 1/3 + 1/5*i + 1/7*j + k
sage: a.reduced_norm()
22826/2205

Python

>>> from sage.all import *
>>> K = QuaternionAlgebra(QQ, -Integer(5), -Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = K._first_ngens(3)
>>> i.reduced_norm()
5
>>> j.reduced_norm()
2
>>> a = Integer(1)/Integer(3) + Integer(1)/Integer(5)*i + Integer(1)/Integer(7)*j + k
>>> a.reduced_norm()
22826/2205

reduced_trace()[source]¶

Return the reduced trace of self.

This is \(2x\) if self is \(x+iy+zj+wk\).

EXAMPLES:

Sage

sage: K.<i,j,k> = QuaternionAlgebra(QQ, -5, -2)
sage: i.reduced_trace()
0
sage: j.reduced_trace()
0
sage: a = 1/3 + 1/5*i + 1/7*j + k
sage: a.reduced_trace()
2/3

Python

>>> from sage.all import *
>>> K = QuaternionAlgebra(QQ, -Integer(5), -Integer(2), names=('i', 'j', 'k',)); (i, j, k,) = K._first_ngens(3)
>>> i.reduced_trace()
0
>>> j.reduced_trace()
0
>>> a = Integer(1)/Integer(3) + Integer(1)/Integer(5)*i + Integer(1)/Integer(7)*j + k
>>> a.reduced_trace()
2/3

sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_generic_v0(*args)[source]¶

EXAMPLES:

Sage

sage: K.<X> = QQ[]
sage: Q.<i,j,k> = QuaternionAlgebra(Frac(K), -5,-19); z = 2/3 + i*X - X^2*j + X^3*k
sage: f, t = z.__reduce__()
sage: sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_generic_v0(*t)
2/3 + X*i + (-X^2)*j + X^3*k
sage: sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_generic_v0(*t) == z
True

Python

>>> from sage.all import *
>>> K = QQ['X']; (X,) = K._first_ngens(1)
>>> Q = QuaternionAlgebra(Frac(K), -Integer(5),-Integer(19), names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3); z = Integer(2)/Integer(3) + i*X - X**Integer(2)*j + X**Integer(3)*k
>>> f, t = z.__reduce__()
>>> sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_generic_v0(*t)
2/3 + X*i + (-X^2)*j + X^3*k
>>> sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_generic_v0(*t) == z
True

sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_number_field_v0(*args)[source]¶

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: K.<a> = QQ[2^(1/3)]; Q.<i,j,k> = QuaternionAlgebra(K, -3, a); z = i + j
sage: f, t = z.__reduce__()
sage: sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_number_field_v0(*t)
i + j
sage: sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_number_field_v0(*t) == z
True

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> K = QQ[Integer(2)**(Integer(1)/Integer(3))]; (a,) = K._first_ngens(1); Q = QuaternionAlgebra(K, -Integer(3), a, names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3); z = i + j
>>> f, t = z.__reduce__()
>>> sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_number_field_v0(*t)
i + j
>>> sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_number_field_v0(*t) == z
True

sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_rational_field_v0(*args)[source]¶

EXAMPLES:

Sage

sage: Q.<i,j,k> = QuaternionAlgebra(-5,-19); a = 2/3 + i*5/7 - j*2/5 +19/2
sage: f, t = a.__reduce__()
sage: sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_rational_field_v0(*t)
61/6 + 5/7*i - 2/5*j

Python

>>> from sage.all import *
>>> Q = QuaternionAlgebra(-Integer(5),-Integer(19), names=('i', 'j', 'k',)); (i, j, k,) = Q._first_ngens(3); a = Integer(2)/Integer(3) + i*Integer(5)/Integer(7) - j*Integer(2)/Integer(5) +Integer(19)/Integer(2)
>>> f, t = a.__reduce__()
>>> sage.algebras.quatalg.quaternion_algebra_element.unpickle_QuaternionAlgebraElement_rational_field_v0(*t)
61/6 + 5/7*i - 2/5*j