#9400: modify the NumberField constructor to pass in optional integer B such
that
all the internal pari routines will replace the discriminant by its gcd
with B, making some things massively faster.
-----------------------------+----------------------------------------------
Reporter: was | Owner: davidloeffler
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.6
Component: number fields | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by was):
{{{
On Tue, Aug 17, 2010 at 5:03 PM, Chan-Ho Kim <chanho.math> wrote:
> Dear William,
> I think that there is a bug on trac 9400 patch.
> My current SAGE is (SAGE 4.5.2 + trac 9400 patch only) in VM.
> When I use `maximize_at_prime,'
>
> K.<a> = NumberField(x^6 + 9*x^5 - 8410*x^4 - 88580*x^3 + 18705368*x^2 +
> 99820416*x - 12230355456, maximize_at_primes=[3]) ; K.primes_above(3)
> this decomposition in K works as you mentioned.
>
> However, in this ``small'' number field
>
> F.<a> = NumberField(x^3 - x^2 - 24*x + 32, maximize_at_primes=[3]) ;
> F.primes_above(3)
> the low precision error occurs if I add `maximize_at_primes=[3].'
That's not a bug in maximize_at_primes or finding the primes above 3.
But it *is* an issue with *printing* the ideals out that it finds over 3.
Evidently, when printing is_principal is called on each ideal currently,
and this leads to a problem. This is not surprising, given that deciding
whether or not an ideal is principal requires knowing the class group in
general, and the equation order of F that you define above is not only
deficient at 3. You need to also maximize at 2. See:
sage: F.<a> = NumberField(x^3 - x^2 - 24*x + 32, maximize_at_primes=[2,3])
sage: F.primes_above(3)
[Fractional ideal (-1/2*a^2 - 3/2*a + 5), Fractional ideal (-1/2*a^2 +
5/2*a - 1)]
So in short, this is not a bug. If you try to compute with number fields
and pass in the maximize_at_primes option, certain things can't possibly
work.
That said, I'm not a big fan of how ideals print. Maybe Jereon's
suggestion -- just *always* print with the PARI 2-element representation
-- is the way to go. That might get around this problem.
> BTW, I also have one more question:
> Can I add `maximize_at_prime=[p]' in `hecke_eigenvalue_field()'?
You'll have to dive in and start hacking at the source code of Sage to do
that....
-- William
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9400#comment:15>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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