# procedure integralpoints produces wrong answer

Elliptic Curve E defined by y^2 = x^3 - 4627*x + 13870 over Rational Field, rank=4

The procedure E.integral_points(verbose=True) uses a MW basis which isn't a basis. Therefore the answer is not correct. There are at least 60 integral points on this curve, but the procedure finds only 58 points. In fact the pair E(1676505139935,±2170737625086802600) is missing. The value of the regulator of the four basis points used in the procedure is 4 times the value of the regulator of four correct basis points. Strange enough the above missing points are linear combinations of the four basis points used in the procedure. I have more examples with similar errors. I used the SAGE app version 6.1.1 on OSX Maverick as well as SAGE version 5.13 in a Virtual Box under Windows 7.

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The additional point is found with Sage version 7.5.rc2 so will be there in the next release. There was a bug fix since 7.4. However the MW basis used is the same in each case, with regulator 5.695... and this is also what mwrank gives, with a claim that the given points are saturated. I also just checked by hand that the generators are OK.

I think that your diagnosis was put off by a misleading message printed with verbose=True: the curve has 2 real components and there are rational points on the non-identity component (the "egg"). In the algorithm, the first step is to replace the full MW basis with a basis for the index 2 subgroup $E(\mathbb{Q})\cap E^0(\mathbb{R}}$, but this is allowed for later.

In fact the bug was quite different, and due to a precision problem which has been fixed.

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This might be the same bug as described by trac tickets 10152 and 12095. Both have been closed as duplicates of trac ticket 10973 where a lot of work is being done on integral points on elliptic curves over number fields.

If you think this is different or if you can help, eg by reviewing the patches, join the discussion at http://trac.sagemath.org/ticket/10973.

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