def normal_form(self, partial=False):
r"""
Return the normal form of this torsion quadratic module.
Two torsion quadratic modules are isomorphic if and only if they have
the same value modules and the same normal form.
A torsion quadratic module `(T,q)` with values in `\QQ/n\ZZ` is
in normal form if the rescaled quadratic module `(T, q/n)`
with values in `\QQ/\ZZ` is in normal form.
For the definition of normal form see [MirMor2009]_ IV Definition 4.6.
Below are some of its properties.
Let `p` be odd and `u` be the smallest non-square modulo `p`.
The normal form is a diagonal matrix with diagonal entries either `p^n`
or `u p^n`.
If `p = 2` is even, then the normal form consists of
1 x 1 blocks of the form
.. MATH::
(0), \quad 2^n(1),\quad 2^n(3),\quad 2^n(5) ,\quad 2^n(7)
or of `2 \times 2` blocks of the form
.. MATH::
2^n
\left(\begin{matrix}
2 & 1\\
1 & 2
\end{matrix}\right), \quad
2^n
\left(\begin{matrix}
0 & 1\\
1 & 0
\end{matrix}\right).
The blocks are ordered by their valuation.
INPUT:
- partial - bool (default: ``False``) return only a partial normal form
it is not unique but still useful to extract invariants
OUTPUT:
- a torsion quadratic module
EXAMPLES::
sage: L1=IntegralLattice(matrix([[-2,0,0],[0,1,0],[0,0,4]]))
sage: L1.discriminant_group().normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2 0]
[ 0 1/4]
sage: L2=IntegralLattice(matrix([[-2,0,0],[0,1,0],[0,0,-4]]))
sage: L2.discriminant_group().normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2 0]
[ 0 1/4]
We check that :trac:`24864` is fixed::
sage: L1=IntegralLattice(matrix([[-4,0,0],[0,4,0],[0,0,-2]]))
sage: AL1=L1.discriminant_group()
sage: L2=IntegralLattice(matrix([[-4,0,0],[0,-4,0],[0,0,2]]))
sage: AL2=L2.discriminant_group()
sage: AL1.normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2 0 0]
[ 0 1/4 0]
[ 0 0 5/4]
sage: AL2.normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2 0 0]
[ 0 1/4 0]
[ 0 0 5/4]
Some exotic cases::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: D4_gram = Matrix(ZZ,4,4,[2,0,0,-1,0,2,0,-1,0,0,2,-1,-1,-1,-1,2])
sage: D4 = FreeQuadraticModule(ZZ,4,D4_gram)
sage: D4dual = D4.span(D4_gram.inverse())
sage: T = TorsionQuadraticModule((1/6)*D4dual,D4)
sage: T
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/18 1/12 5/36 1/36]
[ 1/12 1/6 1/36 1/9]
[ 5/36 1/36 1/36 11/72]
[ 1/36 1/9 11/72 1/36]
sage: T.normal_form()
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/6 1/12 0 0 0 0 0 0]
[1/12 1/6 0 0 0 0 0 0]
[ 0 0 1/12 1/24 0 0 0 0]
[ 0 0 1/24 1/12 0 0 0 0]
[ 0 0 0 0 1/9 0 0 0]
[ 0 0 0 0 0 1/9 0 0]
[ 0 0 0 0 0 0 1/9 0]
[ 0 0 0 0 0 0 0 1/9]
TESTS:
A degenerate case::
sage: T = TorsionQuadraticModule((1/6)*D4dual, D4, modulus=1/36)
sage: T.normal_form()
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/18)Z:
[1/36 1/72 0 0 0 0 0 0]
[1/72 1/36 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
"""
gens = []
from sage.quadratic_forms.genera.normal_form import p_adic_normal_form, _normalize
for p in self.annihilator().gen().prime_divisors():
D_p = self.primary_part(p)
q_p = D_p.gram_matrix_quadratic()
q_p = q_p / D_p._modulus_qf
# continue with the non-degenerate part
r = q_p.rank()
if r != q_p.ncols():
U = q_p._clear_denom()[0].hermite_form(transformation=True)[1]
else:
U = q_p.parent().identity_matrix()
kernel = U[r:, :]
nondeg = U[:r, :]
q_p = nondeg * q_p * nondeg.T
# the normal form is implemented for p-adic lattices
# so we should work with the lattice q_p --> q_p^-1
q_p1 = q_p.inverse()
prec = self.annihilator().gen().valuation(p) + 5
D, U = p_adic_normal_form(q_p1, p, precision=prec + 5, partial=partial)
# if we compute the inverse in the p-adics everything explodes --> go to ZZ
U = U.change_ring(ZZ).inverse().transpose()
# the inverse is in normal form - so to get a normal form for the original one
# it is enough to massage each 1x1 resp. 2x2 block.
U = U.change_ring(Zp(p, type='fixed-mod', prec=prec)).change_ring(ZZ)
D = U * q_p * U.T * p**q_p.denominator().valuation(p)
D = D.change_ring(Zp(p, type='fixed-mod', prec=prec))
_, U1 = _normalize(D, normal_odd=False)
U = U1.change_ring(ZZ) * U
# reattach the degenerate part
nondeg = U * nondeg
U = nondeg.stack(kernel)
# apply U to the generators
n = U.ncols()
gens_p = []
for i in range(n):
g = self.V().zero()
for j in range(n):
g += D_p.gens()[j].lift() * U[i, j]
gens_p.append(g)
gens += gens_p
return self.submodule_with_gens(gens)
def normal_form(self, partial=False):
r"""
Return the normal form of this torsion quadratic module.
Two torsion quadratic modules are isomorphic if and only if they have
the same value modules and the same normal form.
A torsion quadratic module `(T,q)` with values in `\QQ/n\ZZ` is
in normal form if the rescaled quadratic module `(T, q/n)`
with values in `\QQ/\ZZ` is in normal form.
For the definition of normal form see [MirMor2009]_ IV Definition 4.6.
Below are some of its properties.
Let `p` be odd and `u` be the smallest non-square modulo `p`.
The normal form is a diagonal matrix with diagonal entries either `p^n`
or `u p^n`.
If `p = 2` is even, then the normal form consists of
1 x 1 blocks of the form
.. MATH::
(0), \quad 2^n(1),\quad 2^n(3),\quad 2^n(5) ,\quad 2^n(7)
or of `2 \times 2` blocks of the form
.. MATH::
2^n
\left(\begin{matrix}
2 & 1\\
1 & 2
\end{matrix}\right), \quad
2^n
\left(\begin{matrix}
0 & 1\\
1 & 0
\end{matrix}\right).
The blocks are ordered by their valuation.
INPUT:
- partial - bool (default: ``False``) return only a partial normal form
it is not unique but still useful to extract invariants
OUTPUT:
- a torsion quadratic module
EXAMPLES::
sage: L1=IntegralLattice(matrix([[-2,0,0],[0,1,0],[0,0,4]]))
sage: L1.discriminant_group().normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2 0]
[ 0 1/4]
sage: L2=IntegralLattice(matrix([[-2,0,0],[0,1,0],[0,0,-4]]))
sage: L2.discriminant_group().normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2 0]
[ 0 1/4]
We check that :trac:`24864` is fixed::
sage: L1=IntegralLattice(matrix([[-4,0,0],[0,4,0],[0,0,-2]]))
sage: AL1=L1.discriminant_group()
sage: L2=IntegralLattice(matrix([[-4,0,0],[0,-4,0],[0,0,2]]))
sage: AL2=L2.discriminant_group()
sage: AL1.normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2 0 0]
[ 0 1/4 0]
[ 0 0 5/4]
sage: AL2.normal_form()
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2 0 0]
[ 0 1/4 0]
[ 0 0 5/4]
Some exotic cases::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: D4_gram = Matrix(ZZ,4,4,[2,0,0,-1,0,2,0,-1,0,0,2,-1,-1,-1,-1,2])
sage: D4 = FreeQuadraticModule(ZZ,4,D4_gram)
sage: D4dual = D4.span(D4_gram.inverse())
sage: T = TorsionQuadraticModule((1/6)*D4dual,D4)
sage: T
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[1/18 5/36 0 0]
[5/36 1/18 5/36 5/36]
[ 0 5/36 1/36 1/72]
[ 0 5/36 1/72 1/36]
sage: T.normal_form()
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/6 1/12 0 0 0 0 0 0]
[1/12 1/6 0 0 0 0 0 0]
[ 0 0 1/12 1/24 0 0 0 0]
[ 0 0 1/24 1/12 0 0 0 0]
[ 0 0 0 0 1/9 0 0 0]
[ 0 0 0 0 0 1/9 0 0]
[ 0 0 0 0 0 0 1/9 0]
[ 0 0 0 0 0 0 0 1/9]
TESTS:
A degenerate case::
sage: T = TorsionQuadraticModule((1/6)*D4dual, D4, modulus=1/36)
sage: T.normal_form()
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/18)Z:
[1/36 1/72 0 0 0 0 0 0]
[1/72 1/36 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0]
"""
gens = []
from sage.quadratic_forms.genera.normal_form import p_adic_normal_form, _normalize
for p in self.annihilator().gen().prime_divisors():
D_p = self.primary_part(p)
q_p = D_p.gram_matrix_quadratic()
q_p = q_p / D_p._modulus_qf
# continue with the non-degenerate part
r = q_p.rank()
if r != q_p.ncols():
U = q_p._clear_denom()[0].hermite_form(transformation=True)[1]
else:
U = q_p.parent().identity_matrix()
kernel = U[r:,:]
nondeg = U[:r,:]
q_p = nondeg * q_p * nondeg.T
# the normal form is implemented for p-adic lattices
# so we should work with the lattice q_p --> q_p^-1
q_p1 = q_p.inverse()
prec = self.annihilator().gen().valuation(p) + 5
D, U = p_adic_normal_form(q_p1, p, precision=prec + 5, partial=partial)
# if we compute the inverse in the p-adics everything explodes --> go to ZZ
U = U.change_ring(ZZ).inverse().transpose()
# the inverse is in normal form - so to get a normal form for the original one
# it is enough to massage each 1x1 resp. 2x2 block.
U = U.change_ring(Zp(p, type='fixed-mod', prec=prec)).change_ring(ZZ)
D = U * q_p * U.T * p**q_p.denominator().valuation(p)
D = D.change_ring(Zp(p, type='fixed-mod', prec=prec))
_, U1 = _normalize(D, normal_odd=False)
U = U1.change_ring(ZZ) * U
# reattach the degenerate part
nondeg = U * nondeg
U = nondeg.stack(kernel)
#apply U to the generators
n = U.ncols()
gens_p = []
for i in range(n):
g = self.V().zero()
for j in range(n):
g += D_p.gens()[j].lift() * U[i,j]
gens_p.append(g)
gens += gens_p
return self.submodule_with_gens(gens)
