#14472: some elliptic curve functions over number fields fail over relative
fields
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Reporter: cremona | Owner:
cremona
Type: defect | Status: new
Priority: major | Milestone:
sage-5.10
Component: elliptic curves | Resolution:
Keywords: elliptic curve relative number field | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by cremona):
Replying to [comment:1 cremona]:
> Note: after changing ZK.degree() to ZK.absolute_degree() on line 651 of
{{{ell_numberfield.py}}} it still fails, since the code in lines 656--658
(which I wrote, I think) does not work for relative number fields. The
purpose of these lines is to set r,s,t to "least residues" modulo 2,3,2 of
three successive quantities.
Follow-up: the old code for _reduce_model() was flawed, as follows: to
reduce a1,a2,a3 modulo 2,3,2 respectively, it attempted to reduce their
coordinates as given by list(a1), etc. Firstly this fails for relative
extensions, but it is also misguided since there is no reason why the
list() coordinates should be integral. I have changed this to work with
the coordinates with respect to an integral basis, which is good for
relative extensions. Only one small problem: the doctest on line 860
which used to return (as a minimal model over Q(a) where a=sqrt(5))
{{{
(0, 1, 0, a - 33, -2*a + 64)
}}}
but now gives
{{{
(0, -3/2*a - 1/2, 0, 3/2*a - 59/2, 27/2*a + 155/2)
}}}
which does not look so nice. Note that the integral basis here is [1/2*a
+ 1/2, a] and that with respect to this basis 1 has coordinates (2,-1)
while -3/2*a - 1/2 has coordinates (-1,-1) so (counterintuitively) 1 is
not reduced mod 2 but -(3a+1)/2 is!
Note that the integral basis computed does depend on the polynomial used
to generate the field:
{{{
sage: QuadraticField(5,'a').ring_of_integers().gens()
[1/2*a + 1/2, a]
sage: NumberField(x^2-x-1,'a').ring_of_integers().gens()
[1, a]
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14472#comment:2>
Sage <http://www.sagemath.org>
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