def level_ideal(self):
"""
Determines the level of the quadratic form (over R), which is the
smallest ideal N of R such that N * (the matrix of 2*Q)^(-1) is
in R with diagonal in 2*R.
(Caveat: This always returns the principal ideal when working over a field!)
WARNING: THIS ONLY WORKS OVER A PID RING OF INTEGERS FOR NOW!
(Waiting for Sage fractional ideal support.)
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 2, range(1,4))
sage: Q.level_ideal()
Principal ideal (8) of Integer Ring
::
sage: Q1 = QuadraticForm(QQ, 2, range(1,4))
sage: Q1.level_ideal()
Principal ideal (1) of Rational Field
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.level_ideal()
Principal ideal (420) of Integer Ring
"""
##############################################################
## To do this properly, the level should be the inverse of the
## fractional ideal (over R) generated by the entries whose
## denominators we take above. =)
##############################################################
return ideal(self.base_ring()(self.level()))
def level(self):
r"""
Determines the level of the quadratic form over a PID, which is a
generator for the smallest ideal `N` of `R` such that N * (the matrix of
2*Q)^(-1) is in R with diagonal in 2*R.
Over `\ZZ` this returns a non-negative number.
(Caveat: This always returns the unit ideal when working over a field!)
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 2, range(1,4))
sage: Q.level()
8
sage: Q1 = QuadraticForm(QQ, 2, range(1,4))
sage: Q1.level() # random
UserWarning: Warning -- The level of a quadratic form over a field is always 1. Do you really want to do this?!?
1
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.level()
420
"""
## Try to return the cached level
try:
return self.__level
except AttributeError:
## Check that the base ring is a PID
if not is_PrincipalIdealDomain(self.base_ring()):
raise TypeError("Oops! The level (as a number) is only defined over a Principal Ideal Domain. Try using level_ideal().")
## Warn the user if the form is defined over a field!
if self.base_ring().is_field():
warn("Warning -- The level of a quadratic form over a field is always 1. Do you really want to do this?!?")
#raise RuntimeError, "Warning -- The level of a quadratic form over a field is always 1. Do you really want to do this?!?"
## Check invertibility and find the inverse
try:
mat_inv = self.matrix()**(-1)
except ZeroDivisionError:
raise TypeError("Oops! The quadratic form is degenerate (i.e. det = 0). =(")
## Compute the level
inv_denoms = []
for i in range(self.dim()):
for j in range(i, self.dim()):
if (i == j):
inv_denoms += [denominator(mat_inv[i,j] / 2)]
else:
inv_denoms += [denominator(mat_inv[i,j])]
lvl = LCM(inv_denoms)
lvl = ideal(self.base_ring()(lvl)).gen()
##############################################################
## To do this properly, the level should be the inverse of the
## fractional ideal (over R) generated by the entries whose
## denominators we take above. =)
##############################################################
## Normalize the result over ZZ
if self.base_ring() == IntegerRing():
lvl = abs(lvl)
## Cache and return the level
self.__level = lvl
return lvl