#4897: integral_points() misses some points
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Reporter: cremona | Owner: was
Type: defect | Status: new
Priority: major | Milestone: sage-3.4
Component: number theory | Resolution:
Keywords: elliptic curve integral points |
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Comment (by cremona):
It was not that function in the end, but the point_preprocessing function.
{{{
Here's the problem (and it is not in the function I thought it was.
The curve 20160bg2 has two real components. The generators in the
database are
P1=(-18,72) and P2=(-14,0) which are both on the non-identity
component. Preprocessing replaces those by P1+P2, 2*P1 (which
generate a subgroup H of index 2) and the LLL-reduction changes that
to use Q1=P1+P2=(36,-180) and Q2=-P1+P2=(1476,-56700).
The missing point (168,2160) is 2*P1+P2+T where T is the torsion point
(6,0). This is not in H (even up to torsion).
A better way to do the preprocessing here is to take a torsion point
on the egg (e.g. T=(6,0)) and add that to the generators. Similarly,
if there is a torsion point on the egg (necessarily of even order)
then we should add it to any of the generators which are on the egg.
Only if all torsion points are on the identity component should we do
what we currently do.
}}}
I have implemented this change and am currently testing. Patch up later
today!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4897#comment:2>
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