Opened 12 years ago

Closed 11 years ago

# bug in integral_points (for elliptic curves over Q)

Reported by: Owned by: John Cremona John Cremona major sage-duplicate/invalid/wontfix elliptic curves integral points sd32 gaganskehon, Jamie Weigandt William Stein N/A

### Description

```sage: E = EllipticCurve('2082a1')
sage: [P[0] for P in E.integral_points()]
[-11, -2, 4, 13]
```

but

```sage: E.lift_x(507525709)
(507525709 : 11433453531221 : 1)
```

so we miss one point. Note that this point is 13*P where P is the generator, and our code computes the bound to be 12.

John Cannon is sending me a complete list of integral points for all curves in the Cremona database, computed using Magma, using some recent enhancements. Previously I had checked that Sage and Magma agreed to conductor 10000, so their enhancements must also fix bugs.

```Magma V2.15-15    Thu Oct 21 2010 20:56:33 on John-laptop [Seed = 2704111428]
Type ? for help.  Type <Ctrl>-D to quit.

> E:=EllipticCurve([1,0,1,-118,584]);
> IntegralPoints(E);
[ (13 : 29 : 1), (4 : 11 : 1), (-11 : 29 : 1), (-2 : -28 : 1), (507525709 : -11433961056931
: 1) ]
[
[ <(13 : 29 : 1), 1> ],
[ <(13 : 29 : 1), 2> ],
[ <(13 : 29 : 1), 3> ],
[ <(13 : 29 : 1), 4> ],
[ <(13 : 29 : 1), 13> ]
]
19
```

### comment:1 Changed 12 years ago by John Cremona

More data points: with E=2082a1 and using the S_integral_points function, with S=[2] or S=[3] the coefficient bound used is 11, while with S=[2,3] it is 13 and the extra integral point is found! Clearly, a priori, enlarging S should not ever reduce this bound, so something is wrong there. This shows that both integral_points() and S_integral_points() need fixing.

### comment:2 Changed 12 years ago by John Cremona

Here is a list of curves and integral x-coordinates not found by Sage, as provided by John Cannon. It is not necessarily complete, but is a minimal set of test cases for any patch fixing this.

```   Curve        Points found by Magma not in JC tables
--------------------------
2082a1       507525709
6104b1       23036693
8470g1       1681690
8470g2       18888968
13578j1      106398748
16560w2      102492113
23808e1      22259471153
35739g1      148611521
36094c1      1733011457
36600bd1     880839998
38646bd1     633070349
38752c1      71243131
39270x2      15427732395
39330bu2     28736032507
39715e1      367032124
43050c2      127290394
43770f1      247
44282b1      704349106
48294d1      26406701298449
51414v1      649726
51825h1      29581907
52200e1      62128741241070
56355b4      9817312168
56392b1      49071409
60610w2      81996154
60648bm2     63137880         Curves missing around here
63099e1      37015368
65481c2      21083414         Curves missing around here
75730a1      3786617429
76128p1      50684407
78078by1     1361015 5344835475
83490r2      87561186
90870a2      72592777212
95337a1      167959
100096h1     3318161
101320f2     14134787
103520a1     94894808
108990cb2    71041501977
109330k2     377482949
109968f1     135059618524
110976h2     116239257
113706i1     1945 9319420657
114240ev1    938125
115920bb2    184673663
116088n1     1374364421
116511b1     1375325
124110y1     37815783
129420b1     1172841435
```

### comment:5 Changed 12 years ago by Justin Walker

See #10973. The patch there finds the missing integral points mentioned here.

### comment:6 Changed 11 years ago by Alyson Deines

This is fixed by #10973. It should be closed as duplicate.

### comment:7 Changed 11 years ago by William Stein

Resolution: → duplicate new → closed

### comment:8 Changed 11 years ago by William Stein

Work issues: → sd32

### comment:9 Changed 11 years ago by Minh Van Nguyen

Milestone: sage-4.7.2 → sage-duplicate/invalid/wontfix

### comment:10 Changed 11 years ago by Leif Leonhardy

Keywords: sd32 added → William Stein sd32
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