def brown_invariant(self):
r"""
Return the Brown invariant of this torsion quadratic form.
Let `(D,q)` be a torsion quadratic module with values in `\QQ / 2 \ZZ`.
The Brown invariant `Br(D,q) \in \Zmod{8}` is defined by the equation
.. MATH::
\exp \left( \frac{2 \pi i }{8} Br(q)\right) =
\frac{1}{\sqrt{D}} \sum_{x \in D} \exp(i \pi q(x)).
The Brown invariant is additive with respect to direct sums of
torsion quadratic modules.
OUTPUT:
- an element of `\Zmod{8}`
EXAMPLES::
sage: L = IntegralLattice("D4")
sage: D = L.discriminant_group()
sage: D.brown_invariant()
4
We require the quadratic form to be defined modulo `2 \ZZ`::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ,3,matrix.identity(3))
sage: T = TorsionQuadraticModule((1/10)*V, V)
sage: T.brown_invariant()
Traceback (most recent call last):
...
ValueError: the torsion quadratic form must have values in QQ / 2 ZZ
"""
if self._modulus_qf != 2:
raise ValueError("the torsion quadratic form must have values in "
"QQ / 2 ZZ")
from sage.quadratic_forms.genera.normal_form import collect_small_blocks
brown = IntegerModRing(8).zero()
for p in self.annihilator().gen().prime_divisors():
q = self.primary_part(p).normal_form()
q = q.gram_matrix_quadratic()
L = collect_small_blocks(q)
for qi in L:
brown += _brown_indecomposable(qi, p)
return brown
def brown_invariant(self):
r"""
Return the Brown invariant of this torsion quadratic form.
Let `(D,q)` be a torsion quadratic module with values in `\QQ / 2 \ZZ`.
The Brown invariant `Br(D,q) \in \Zmod{8}` is defined by the equation
.. MATH::
\exp \left( \frac{2 \pi i }{8} Br(q)\right) =
\frac{1}{\sqrt{D}} \sum_{x \in D} \exp(i \pi q(x)).
The Brown invariant is additive with respect to direct sums of
torsion quadratic modules.
OUTPUT:
- an element of `\Zmod{8}`
EXAMPLES::
sage: L = IntegralLattice("D4")
sage: D = L.discriminant_group()
sage: D.brown_invariant()
4
We require the quadratic form to be defined modulo `2 \ZZ`::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ,3,matrix.identity(3))
sage: T = TorsionQuadraticModule((1/10)*V, V)
sage: T.brown_invariant()
Traceback (most recent call last):
...
ValueError: The torsion quadratic form must have values in\QQ / 2\ZZ
"""
if self._modulus_qf != 2:
raise ValueError("The torsion quadratic form must have values in"
"\QQ / 2\ZZ")
from sage.quadratic_forms.genera.normal_form import collect_small_blocks
brown = IntegerModRing(8).zero()
for p in self.annihilator().gen().prime_divisors():
q = self.primary_part(p).normal_form()
q = q.gram_matrix_quadratic()
L = collect_small_blocks(q)
for qi in L:
brown += _brown_indecomposable(qi,p)
return brown