#5307: Bug in conductor() over number fields
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Reporter: cremona | Owner: was
Type: defect | Status: new
Priority: major | Milestone: sage-3.4
Component: number theory | Keywords: elliptic curve
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Comment(by cremona):
Diagnosis of the problem, while lies in the implementation of Tate's
algorithm at a prime ideal P when P is not principal: we use a
uniformiser pi of P, but we use it in two different ways. First, there
are various places where integers (of the field) which are known to have
valuation at least i are divided by {{{pi^i}}}. Here, in order to keep
everything integral, we use a uniformizer computed via K.uniformizer(P,
'negative'), which has valuation 1 at P (of course) and nagative or zero
valuation elsewhere. But there is a second way in which pi is used: in
computing the appropriate [u,r,s,t]-transforms. For example, in one place
we need an r-transform where r is 0 mod P but something specific mod
{{{P^2}}}; so we write r=r0*pi and compute r0 mod P and then multiply by
pi. But now, it matters if pi is not integral!
The solution I came up with was to compute two uniformisers, one (pi) as
above and another (called ipi) integral at all primes. I use the
appropriate one in the appropriate places.
I made a patch to implement this, and the example above works fine
(doctest added to conductor() in ell_number_field.py).
__But__ I think this needs to be looked at more carefully; while it is no
worse than before (and no different at all at principal primes) I don't
think it is quite right yet.
NB Magma has essentially the same code (I wrote it) but is not fussy about
integrality at all since it does not give local minimal models.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5307#comment:1>
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