#12811: (reduced) norm of quaternion ideals is broken
-----------------------+----------------------------------------------------
Reporter: daniels | Owner: AlexGhitza
Type: defect | Status: new
Priority: major | Milestone: sage-5.0
Component: algebra | Keywords: quaternion, ideal
Work issues: | Report Upstream: N/A
Reviewers: | Authors:
Merged in: | Dependencies:
Stopgaps: |
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1.) The norm of an ideal in quaternion algebra is off by a factor of 2
{{{
sage: A.<i,j,k> = QuaternionAlgebra(-1,-1)
sage: I = A.ideal([i,j,k,1/2 + 1/2*i + 1/2*j + 1/2*k]) # this is
actually an order, so has reduced norm 1
sage: I.left_order()
sage: I.norm()
2
}}}
The reason is that the norm is computed from d(I)=n(I)^2^ d(R) where R is
the left- or right-order of I and d(I) = \sqrt{ abs{det(tr(x_i
\conj{x_j}))} } for a ZZ-basis (x_i) of I. But the matrix returned by
gram_matrix() that is actually used in the computation has as entries
2*tr(x_i \conj{x_j}), thus the determinant is off by a factor of 16 and
the resulting reduced norm off by a factor of 2
2.) Omitting the line 'I.left_order()' in the previous example gives a
RuntimeError:
{{{
sage: A.<i,j,k> = QuaternionAlgebra(-1,-1)
sage: I = A.ideal([i,j,k,1/2 + 1/2*i + 1/2*j + 1/2*k])
sage: I.norm()
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call
last)
/home/daniel/<ipython console> in <module>()
/opt/sage-4.8-linux-64bit-
ubuntu_10.04.3_lts-x86_64-Linux/local/lib/python2.6/site-
packages/sage/algebras/quatalg/quaternion_algebra.pyc in norm(self)
1973 assert r.is_square(), "first is bad!"
1974 r = r.sqrt()
-> 1975 r/= self.quaternion_order().discriminant()
1976 assert r.is_square(), "second is bad!"
1977 return r.sqrt()
/opt/sage-4.8-linux-64bit-
ubuntu_10.04.3_lts-x86_64-Linux/local/lib/python2.6/site-
packages/sage/algebras/quatalg/quaternion_algebra.pyc in
quaternion_order(self)
1692 A = self.__right_order
1693 else:
-> 1694 raise RuntimeError, "unable to determine quaternion
order of ideal without known order"
1695 self.__quaternion_order = A
1696 return A
RuntimeError: unable to determine quaternion order of ideal without known
order
}}}
This is because neither the left- nor the right-order of I have been
computed at this point. There is however no mathematical reason why this
should fail.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12811>
Sage <http://www.sagemath.org>
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