#10017: reduced_basis for number field multiples wrong
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Reporter: schilly | Owner: davidloeffler
Type: defect | Status: new
Priority: major | Milestone: sage-4.6
Component: number fields | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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This was reported by the "report a problem" link:
The reduced_basis for number fields applies LLL to basis of the maximal
order in order to get a reduced basis. The problem is that after applying
LLL, it multiples the transformation matrix by the vector [1,a,a^2,...],
where 'a' is generator of the field - instead of using the vector it
actually used for the maximal order.
How to reproduce:
{{{
x = QQ['x'].0
k = NumberField(x^6 + 2218926655879913714112*x^4 - \
32507675650290949030789018433536*x^3 + \
4923635504174417014460581055002374467948544*x^2 - \
36066074010564497464129951249279114076897746988630016*x + \
264187244046129768986806800244258952598300346857154900812365824,'a')
new_basis = k.reduced_basis()
print new_basis[0].minpoly()
}}}
This prints a crazy big polynomial. Looking a bit at what is returned it
is clear that reduced_basis is multiplying the transformation matrix by
the wrong vector.
To solve you just need to multiply by the vector of generators of the
maximal order:
{{{
mp = pari( k.Minkowski_embedding(k.maximal_order().gens(), prec=120) )
ml = sage.libs.pari.gen.gen.qflll(mp)
T = matrix([[ZZ(y) for y in list(x)] for x in list(ml)]).transpose()
r = Matrix(O.gens())* T
for rr in r[0]:
print rr.minpoly()
}}}
This works fine.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10017>
Sage <http://www.sagemath.org>
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