Opened 10 years ago

Closed 9 years ago

# integral_points missing solutions

Reported by: Owned by: dsm cremona major sage-duplicate/invalid/wontfix elliptic curves Travis Scrimshaw N/A

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.

### comment:1 Changed 10 years ago by dsm

• Description modified (diff)

### comment:2 Changed 10 years ago by dsm

• Type changed from PLEASE CHANGE to defect

### comment:3 Changed 10 years ago by cremona

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.

### comment:4 Changed 10 years ago by dsm

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.

### comment:5 Changed 10 years ago by cremona

• Milestone changed from sage-4.8 to sage-duplicate/invalid/wontfix

### comment:6 Changed 10 years ago by cremona

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.

### comment:7 Changed 9 years ago by tscrim

• Status changed from new to needs_review

### comment:8 Changed 9 years ago by tscrim

• Reviewers set to Travis Scrimshaw
• Status changed from needs_review to positive_review

### comment:9 Changed 9 years ago by jdemeyer

• Resolution set to duplicate
• Status changed from positive_review to closed
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