#4741: [with patch, needs review] Implement S-integral point finding for
elliptic
curves over Q
---------------------------+------------------------------------------------
Reporter: cremona | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-3.3
Component: number theory | Resolution:
Keywords: |
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Changes (by cremona):
* summary: [with patch, not yet ready for review] Implement S-integral
point finding for elliptic curves over Q =>
[with patch, needs review] Implement S-integral
point finding for elliptic curves over Q
Comment:
The third patch fixes a bug (missed a 2-integral points on 37a1) and makes
{{{S_integral_x_coords_with_abs_bounded_by()}}} slightly more efficient,
but we are way slower than Magma and I think that the main culprits are
that function and {{{S_integral_points_with_bounded_mw_coeffs()}}}. In
the integral points case we made the latter much faster by using real
points (since you can tell if a real number is approximately integral) but
th analogous thing for S-integral points is more complicated. Rather than
wait, as we are getting the right points (after substantial testing
against Magma, for example), I would like this to go in now and we can
look into making things faster later.
Details of what is in the 3rd patch:
1. p-adic elliptic log: default precision now 20 not 100, which works
fine. Also converts points to p-adic earlier which avoids dealing with
large rationals when the multiplier factor is needed.
2. Calls to p-adic log from S_integral_points() now multiply by the
necessary factor after taking logs instead of before: it is a
homomorphism, and again this avoinds dealing with larger rationals.
3. Correct use of proof flag (though it is rather crude that when
proof==False we just omit the time-consuming call to
S_integral_x_coords_with_abs_bounded_by(abs_bound)).
4. Some speed improvements to
S_integral_x_coords_with_abs_bounded_by(), and improved comments including
some TODOs.
[Note to Tobias and Michael M: this patch includes the changes in the
patch you sent me.]
I have a timing comparison table and will include it here once I have
worked out how to do wiki tables.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4741#comment:2>
Sage <http://sagemath.org/>
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