Dear Costas,
Thanks for the bug report. As Nils pointed out there are known bugs
in the integral point code which cause solutions to be missed. A lot
of work has been spent on improving this, in part by me, and the main
reason the fixes have not yet been approved and merged is that I still
had more work to do on it and it got forgotten.
This example is odd though. The rank is 3 (as you can see from
E1.gens()) and when you run E1.integral_points(verbose=True) you can
see that it does correctly compute a bound of 3 on the coefficients
of the integral points as a Z-linear combination of the generators.
(There is no torsion). This may be a precision issue since the final
search does use real arithmetic, in the (non-global) function
integral_points_with_bounded_mw_coefficients(). There is a comment in
there which says
# Otherwise it is very very much faster to first compute
# the linear combinations over RR, and only compute them as
# rational points if they are approximately integral.
# Note: making eps larger here will dramatically increase
# the running time. If evidence arises that integral
# points are being missed, it would be better to increase
# the real precision than to increase eps.
and I think you have just found such evidence.
John Cremona
On 14 December 2016 at 17:07, <[email protected]> wrote:
>
> Hi all,
>
> I came across the following example...
>
> sage: E1=EllipticCurve([0,0,0,49,-64]);E1;S=E1.integral_points();S;
>
> Elliptic Curve defined by y^2 = x^3 + 49*x - 64 over Rational Field
> [(4 : 14 : 1), (464 : 9996 : 1)]
>
> but the following integer point Q belongs also to the curve
>
> sage:Q=E1(482042404,10583464697386);Q
>
>
> (482042404 : 10583464697386 : 1)
>
> further, Q=3*M.
>
>
> Any idea why sage did not provide the point Q?
> I used sage version 6.9.
>
> Thank you
>
> Costas
>
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