def global_genus_symbol(self):
"""
Returns the genus of a two times a quadratic form over ZZ. These
are defined by a collection of local genus symbols (a la Chapter
15 of Conway-Sloane), and a signature.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
sage: Q.global_genus_symbol()
Genus of [2 0 0 0]
[0 4 0 0]
[0 0 6 0]
[0 0 0 8]
::
sage: Q = QuadraticForm(ZZ, 4, range(10))
sage: Q.global_genus_symbol()
Genus of [ 0 1 2 3]
[ 1 8 5 6]
[ 2 5 14 8]
[ 3 6 8 18]
"""
## Check that the form is defined over ZZ
if not self.base_ring() == IntegerRing():
raise TypeError, "Oops! The quadratic form is not defined over the integers."
## Return the result
try:
return Genus(self.Hessian_matrix())
except Exception:
raise TypeError, "Oops! There is a problem computing the genus symbols for this form."
def genus(self):
r"""
Return the genus of this lattice.
EXAMPLES::
sage: L = IntegralLattice("U")
sage: L.genus()
Genus of
[0 1]
[1 0]
Genus symbol at 2: 1^2
"""
from sage.quadratic_forms.genera.genus import Genus
return Genus(self.gram_matrix())
def genus(self):
r"""
Return the genus of this lattice.
EXAMPLES::
sage: from sage.modules.free_quadratic_module_integer_symmetric import IntegralLattice
sage: L = IntegralLattice(Matrix(ZZ,2,2,[0,1,1,0]))
sage: L.genus()
Genus of
[0 1]
[1 0]
Genus symbol at 2: 1^2
"""
from sage.quadratic_forms.genera.genus import Genus
return Genus(self.gram_matrix())
def global_genus_symbol(self):
r"""
Return the genus of two times a quadratic form over `\ZZ`.
These are defined by a collection of local genus symbols (a la
Chapter 15 of Conway-Sloane [CS1999]_), and a signature.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
sage: Q.global_genus_symbol()
Genus of
[2 0 0 0]
[0 4 0 0]
[0 0 6 0]
[0 0 0 8]
Signature: (4, 0)
Genus symbol at 2: [2^-2 4^1 8^1]_6
Genus symbol at 3: 1^3 3^-1
::
sage: Q = QuadraticForm(ZZ, 4, range(10))
sage: Q.global_genus_symbol()
Genus of
[ 0 1 2 3]
[ 1 8 5 6]
[ 2 5 14 8]
[ 3 6 8 18]
Signature: (3, 1)
Genus symbol at 2: 1^-4
Genus symbol at 563: 1^3 563^-1
"""
if self.base_ring() is not ZZ:
raise TypeError("the quadratic form is not defined over the integers")
return Genus(self.Hessian_matrix())