integral_points missing solutions

[sagetrac-dsm]

As reported on sage-support, integral_points can miss solutions:

sage: E=EllipticCurve([0,-82569375])
sage: E.integral_points(verbose=True)
Using mw_basis  [(450 : 2925 : 1), (900 : 25425 : 1), (1800 : 75825 : 1), (4600/9 : 192725/27 : 1)]
e1,e2,e3:  -217.725687258292 - 377.111952444214*I -217.725687258292 + 377.111952444214*I 435.451374516585
Minimal eigenvalue of height pairing matrix:  2.03352484556965
x-coords of points on non-compact component with  436 <=x<= 870
[436, 450, 666]
starting search of remaining points using coefficient bound  4
x-coords of extra integral points:
[436, 450, 666, 900, 1150, 1800, 2619, 26154, 27675]
Total number of integral points: 9
[(436 : 559 : 1), (450 : 2925 : 1), (666 : 14589 : 1), (900 : 25425 : 1), (1150 : 37925 : 1), (1800 : 75825 : 1), (2619 : 133722 : 1), (26154 : 4229667 : 1), (27675 : 4603950 : 1)]

but

sage: 20477027135825**2==748476100**3-82569375
True

The (only?) missing solution can be found by increasing the precision in integral_points_with_bounded_mw_coeffs from 100 to 120:

[(436 : 559 : 1), (450 : 2925 : 1), (666 : 14589 : 1), (900 : 25425 : 1), (1150 : 37925 : 1), (1800 : 75825 : 1), (2619 : 133722 : 1), (26154 : 4229667 : 1), (27675 : 4603950 : 1), (748476100 : 20477027135825 : 1)]

but obviously this isn't robust. With a bit of tweaking (and maybe some RealIntervalFieldElements) we could ensure that we use sufficient precision without hardcoding an arbitrary parameter.

Component: elliptic curves

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/12095

[sagetrac-dsm]

Description changed:

--- 
+++ 
@@ -15,10 +15,12 @@
 Total number of integral points: 9
 [(436 : 559 : 1), (450 : 2925 : 1), (666 : 14589 : 1), (900 : 25425 : 1), (1150 : 37925 : 1), (1800 : 75825 : 1), (2619 : 133722 : 1), (26154 : 4229667 : 1), (27675 : 4603950 : 1)]
 ```
-but {{{
+but 
+
+```
 sage: 20477027135825**2==748476100**3-82569375
 True
-}}}
+```
 
 The (only?) missing solution can be found by increasing the precision in integral_points_with_bounded_mw_coeffs from 100 to 120:
 

[JohnCremona]

comment:3

Please see #10973: I don't think this needs a separate ticket. But it is a very useful observation to make. There are still issues at #10973.

[sagetrac-dsm]

comment:4

Not sure how I missed that-- it shows up in the first page of searches! Maybe I assumed it was only about generalization to other number fields?

Anyway, we can close this as a duplicate. The latest patch there fixes this case, though I still think hardcoding the precision is asking for trouble.

[JohnCremona]

comment:6

I marked this as a duplicate and put in a cross-reference at #10973.

I agree strongly with your comment about hard-coding precision in that function, for which I take 100% of the blame.

[tscrim]

Reviewer: Travis Scrimshaw