def __init__(self,
V,
W,
gens=None,
modulus=None,
modulus_qf=None,
check=True):
r"""
Initialize ``self``.
TESTS::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: T = TorsionQuadraticModule(ZZ^3, 6*ZZ^3)
sage: TestSuite(T).run()
"""
if check:
if V.rank() != W.rank():
raise ValueError("modules must be of the same rank")
if V.base_ring() is not ZZ:
raise NotImplementedError("only currently implemented over ZZ")
if V.inner_product_matrix() != V.inner_product_matrix().transpose(
):
raise ValueError(
"the cover must have a symmetric inner product")
if gens is not None and V.span(gens) + W != V:
raise ValueError("provided gens do not generate the quotient")
FGP_Module_class.__init__(self, V, W, check=check)
if gens is not None:
self._gens_user = tuple(self(v) for v in gens)
else:
# this is taken care of in the .gens method
# we do not want this at initialization
self._gens_user = None
# compute the modulus - this may be expensive
if modulus is None or check:
# The inner product of two elements `b(v1+W,v2+W)`
# is defined `mod (V,W)`
num = V.basis_matrix() * V.inner_product_matrix() * W.basis_matrix(
).T
self._modulus = gcd(num.list())
if modulus is not None:
if check and self._modulus / modulus not in self.base_ring():
raise ValueError("the modulus must divide (V, W)")
self._modulus = modulus
if modulus_qf is None or check:
# The quadratic_product of an element `q(v+W)` is defined
# `\mod 2(V,W) + ZZ\{ (w,w) | w in w\}`
norm = gcd(self.W().gram_matrix().diagonal())
self._modulus_qf = gcd(norm, 2 * self._modulus)
if modulus_qf is not None:
if check and self._modulus_qf / modulus_qf not in self.base_ring():
raise ValueError("the modulus_qf must divide (V, W)")
self._modulus_qf = modulus_qf
def __init__(self, cover, relations):
r"""
EXAMPLES::
sage: G = AdditiveAbelianGroup([0]); G # indirect doctest
Additive abelian group isomorphic to Z
sage: G == loads(dumps(G))
True
"""
FGP_Module_class.__init__(self, cover, relations)
def __init__(self, cover, relations):
r"""
EXAMPLE::
sage: G = AdditiveAbelianGroup([0]); G # indirect doctest
Additive abelian group isomorphic to Z
sage: G == loads(dumps(G))
True
"""
FGP_Module_class.__init__(self, cover, relations)
def __init__(self, V, W, gens=None, modulus=None, modulus_qf=None, check=True):
r"""
Initialize ``self``.
TESTS::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: T = TorsionQuadraticModule(ZZ^3, 6*ZZ^3)
sage: TestSuite(T).run()
"""
if check:
if V.rank() != W.rank():
raise ValueError("modules must be of the same rank")
if V.base_ring() is not ZZ:
raise NotImplementedError("only currently implemented over ZZ")
if V.inner_product_matrix() != V.inner_product_matrix().transpose():
raise ValueError("the cover must have a symmetric inner product")
if gens is not None and V.span(gens) + W != V:
raise ValueError("provided gens do not generate the quotient")
FGP_Module_class.__init__(self, V, W, check=check)
if gens is None:
self._gens = FGP_Module_class.gens(self)
else:
self._gens = [self(v) for v in gens]
if modulus is not None:
if check:
# The inner product of two elements `b(v1+W,v2+W)` is defined `mod (V,W)`
num = gcd([x.inner_product(y) for x in V.gens()
for y in W.gens()])
if num / modulus not in self.base_ring():
raise ValueError("the modulus must divide (V, W)")
self._modulus = modulus
else:
# The inner product of two elements `b(v1+W,v2+W)` is defined `mod (V,W)`
self._modulus = gcd([x.inner_product(y) for x in V.gens()
for y in W.gens()])
if modulus_qf is not None:
if check:
# The quadratic_product of an element `q(v+W)` is defined
# `\mod 2(V,W) + ZZ\{ (w,w) | w in w\}`
norm = gcd(self.W().gram_matrix().diagonal())
num = gcd(norm, 2 * self._modulus)
if num / modulus_qf not in self.base_ring():
raise ValueError("the modulus_qf must divide (V, W)")
self._modulus_qf = modulus_qf
else:
# The quadratic_product of an element `q(v+W)` is defined
# `\mod 2(V,W) + ZZ\{ (w,w) | w in w\}`
norm = gcd(self.W().gram_matrix().diagonal())
self._modulus_qf = gcd(norm, 2 * self._modulus)
def __init__(self, V, W, gens, modulus, modulus_qf):
r"""
Initialize ``self``.
TESTS::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: T = TorsionQuadraticModule(ZZ^3, 6*ZZ^3)
sage: TestSuite(T).run()
"""
FGP_Module_class.__init__(self, V, W, check=True)
if gens is not None:
self._gens_user = tuple(self(g) for g in gens)
else:
# this is taken care of in the .gens method
# we do not want this at initialization
self._gens_user = None
self._modulus = modulus
self._modulus_qf = modulus_qf
def submodule(self, x):
r"""
Return the submodule defined by ``x``.
The modulus of the inner product is inherited from ``self``.
INPUT:
- ``x`` -- list, tuple, or FGP module
OUTPUT:
- a :class:`TorsionQuadraticModule`
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ,3,matrix.identity(3)*5)
sage: T = TorsionQuadraticModule((1/5)*V, V)
sage: T
Finite quadratic module over Integer Ring with invariants (5, 5, 5)
Gram matrix of the quadratic form with values in Q/Z:
[1/5 0 0]
[ 0 1/5 0]
[ 0 0 1/5]
sage: T.submodule(T.gens()[:2])
Finite quadratic module over Integer Ring with invariants (5, 5)
Gram matrix of the quadratic form with values in Q/Z:
[1/5 0]
[ 0 1/5]
"""
T = FGP_Module_class.submodule(self, x)
# We need to explicitly set the _modulus and _modulus_qf
# else the modulus might increase.
T._modulus = self._modulus
T._modulus_qf = self._modulus_qf
return T
def submodule(self, x):
r"""
Return the submodule defined by ``x``.
The modulus of the inner product is inherited from ``self``.
INPUT:
- ``x`` -- list, tuple, or FGP module
OUTPUT:
- a :class:`TorsionQuadraticModule`
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ,3,matrix.identity(3)*5)
sage: T = TorsionQuadraticModule((1/5)*V, V)
sage: T
Finite quadratic module over Integer Ring with invariants (5, 5, 5)
Gram matrix of the quadratic form with values in Q/Z:
[1/5 0 0]
[ 0 1/5 0]
[ 0 0 1/5]
sage: T.submodule(T.gens()[:2])
Finite quadratic module over Integer Ring with invariants (5, 5)
Gram matrix of the quadratic form with values in Q/Z:
[1/5 0]
[ 0 1/5]
"""
T = FGP_Module_class.submodule(self, x)
# We need to explicitly set the _modulus and _modulus_qf
# else the modulus might increase.
T._modulus = self._modulus
T._modulus_qf = self._modulus_qf
return T
def __init__(self,
V,
W,
gens=None,
modulus=None,
modulus_qf=None,
check=True):
r"""
Initialize ``self``.
TESTS::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: T = TorsionQuadraticModule(ZZ^3, 6*ZZ^3)
sage: TestSuite(T).run()
"""
if check:
if V.rank() != W.rank():
raise ValueError("modules must be of the same rank")
if V.base_ring() is not ZZ:
raise NotImplementedError("only currently implemented over ZZ")
if V.inner_product_matrix() != V.inner_product_matrix().transpose(
):
raise ValueError(
"the cover must have a symmetric inner product")
if gens is not None and V.span(gens) + W != V:
raise ValueError("provided gens do not generate the quotient")
FGP_Module_class.__init__(self, V, W, check=check)
if gens is None:
self._gens = FGP_Module_class.gens(self)
else:
self._gens = [self(v) for v in gens]
if modulus is not None:
if check:
# The inner product of two elements `b(v1+W,v2+W)` is defined `mod (V,W)`
num = gcd(
[x.inner_product(y) for x in V.gens() for y in W.gens()])
if num / modulus not in self.base_ring():
raise ValueError("the modulus must divide (V, W)")
self._modulus = modulus
else:
# The inner product of two elements `b(v1+W,v2+W)` is defined `mod (V,W)`
self._modulus = gcd(
[x.inner_product(y) for x in V.gens() for y in W.gens()])
if modulus_qf is not None:
if check:
# The quadratic_product of an element `q(v+W)` is defined
# `\mod 2(V,W) + ZZ\{ (w,w) | w in w\}`
norm = gcd(self.W().gram_matrix().diagonal())
num = gcd(norm, 2 * self._modulus)
if num / modulus_qf not in self.base_ring():
raise ValueError("the modulus_qf must divide (V, W)")
self._modulus_qf = modulus_qf
else:
# The quadratic_product of an element `q(v+W)` is defined
# `\mod 2(V,W) + ZZ\{ (w,w) | w in w\}`
norm = gcd(self.W().gram_matrix().diagonal())
self._modulus_qf = gcd(norm, 2 * self._modulus)