The following table lists the 54 cases known to me of pairs
(E,P) where E is an elliptic curve
over Q and P is a nontorsion point
on E of canonical height h<1/100.
The curves are listed in order of increasing h.
For each curve we list the conductor N,
the coordinates (A,u) of (E,P)
[explicit formulas below]
the coefficients
(a_{1}, a_{2}, a_{3})
(where E is
Y^{2}
+ a_{1} XY
+ a_{3} Y
= X^{3}
+ a_{2} X^{2}
and P is (0,0)),
and the positive integers n<=50 such that nP
is an integral point on the minimal model of E,
followed by the count of such n. These were found
by an exhaustive search over (A,u);
William Stein confirms that the list includes all 32 instances
of N<12000, using the extended Cremona tables.
This list also contains the highest known integral multiple
(n=31, at #10), the largest counts of integral multiples
(16, at #1 and #2), and the largest known ratio
log|D|/h
(11822.2+, at #44; the next largest is 9388.9+ at #1).
As for the most consecutive integral multiples,
the table lists several instances of 12, and in fact
I can prove that there are infinitely many cases where nP
is integral for each n=1,2,...,12; but that is not the record:
there are several 13's, and one 14.
The choice (A,u)=(-45/41,-12/5)
yields a curve E
of conductor 1029210 with coefficients
(a_{1}, a_{2}, a_{3})
= (209,23520,2446080) and a point P=(0,0) whose height of
0.01017273 places it just beyond our table's range but whose first
14 multiples (and also the 18th) are integral points on E.
# | h | N | (A,u) | (a_{1}, a_{2}, a_{3}) | n | |
1 | 0.00445716 | 3990 | (1/2, 3/7) | [253, 5320, 1197000] | 1-7, 9, 10, 12, 14, 15, 18-21 (16) | |
2 | 0.00451934 | 3630 | (-5/8, -1/3) | [-77, -2640, -7920] | 1-9, 12-16, 18, 21 (16) | |
3 | 0.00486993 | 1430 | (1, 1/2) | [-73, 440, -28600] | 1-6, 8-10, 12, 15-18 (14) | |
4 | 0.00498205 | 1470 | (-4/7, -6/5) | [7, 70, -210] | 1-7, 9-12, 14, 15, 18, 25 (15) | |
5 | 0.00563876 | 280 | (1, -2/5) | [-8, -70, -140] | 1-6, 9-13, 15 (12) | |
6 | 0.00571551 | 3822 | (-4/9, -2) | [-133, 546, -49686] | 1-8, 10, 12, 13, 15, 16, 19, 20 (15) | |
7 | 0.00577779 | 6510 | (-2/3, -2) | [11, 210, 1050] | 1-10, 12, 13, 15, 17, 21 (15) | |
8 | 0.00592549 | 1518 | (-2, -6/11) | [-65, -1518, -1518] | 1-6, 9, 10, 12, 15, 18, 20 (12) | |
9 | 0.00603438 | 1890 | (1/2, 1) | [-39, 120, -4200] | 1-8, 10, 12-15 (13) | |
10 | 0.00612648 | 1830 | (-7/13, -3/5) | [-17, -30, 960] | 1-11, 14, 15, 17, 31 (15) | |
11 | 0.00642340 | 350 | (-2, 2) | [-5, -10, 70] | 1-6, 8-10, 12, 13, 18 (12) | |
12 | 0.00656900 | 220110 | (-4/5, 2) | [349, 7590, 2421210] | 1-6, 8-10, 12, 15, 18, 20 (13) | |
13 | 0.00662175 | 6090 | (1, 1) | [-53, 210, -10080] | 1-9, 12-15 (13) | |
14 | 0.00670838 | 735 | (1, -1/2) | [14, -105, 105] | 1-4, 6-9, 11-13, 24 (12) | |
15 | 0.00678178 | 37950 | (1/5, 3/11) | [535, 22770, 10929600] | 1-6, 8-12, 15, 23, 24 (14) | |
16 | 0.00682239 | 1806 | (4, 2) | [-11, -378, -2646] | 1-8, 10-12, 14, 15, 20 (14) | |
17 | 0.00691756 | 7854 | (-9/13, -12/7) | [23, 924, 1848] | 1-6, 9-13, 15, 17 (13) | |
18 | 0.00741674 | 61050 | (-4/5, -2/11) | [-65, -12210, -549450] | 1-6, 8-10, 12, 15, 16, 18 (13) | |
19 | 0.00754335 | 550290 | (-4, 2) | [-211, 6630, -537030] | 1-9, 11-13, 15, 16, 21 (15) | |
20 | 0.00771256 | 11730 | (-20/13, -14/23) | [-133, -7038, -35190] | 1-6, 9-11, 15, 20 (11) | |
21 | 0.00788260 | 14910 | (1/2, -3/5) | [-61, -840, 840] | 1-10, 12, 14, 15 (13) | |
22 | 0.00789904 | 14430 | (4/17, -6) | [29, -450, -17550] | 1-8, 10-12, 14, 17 (13) | |
23 | 0.00791669 | 600 | (-1/3, 2) | [-20, 30, -540] | 1-7, 9-12, 21 (12) | |
24 | 0.00812577 | 4620 | (4/17, -7/22) | [-28, -3300, -89100] | 1-5, 7, 9-11, 13-15 (12) | |
25 | 0.00814044 | 140790 | (-12/23, -2/3) | [-157, -1170, 288990] | 1-8, 10, 12-14, 17, 20 (14) | |
26 | 0.00816727 | 17490 | (-20/41, -14/11) | [-103, 330, -26730] | 1-7, 9-11, 13, 15, 20 (13) | |
27 | 0.00831871 | 7098 | (-3, 3/2) | [-37, 168, -2184] | 1-6, 8-12, 16 (12) | |
28 | 0.00835139 | 2370 | (1, 2) | [-19, 30, -270] | 1-9, 11, 12, 17 (12) | |
29 | 0.00848309 | 95370 | (-4/7, -2/3) | [-103, 990, -84150] | 1-10, 12, 14, 15 (13) | |
30 | 0.00850139 | 7770 | (-5, -4/7) | [-41, -630, -630] | 1-7, 9-11, 13, 15 (12) | |
31 | 0.00855560 | 4474470 | (1/2, -1/3) | [-103, -32760, -2522520] | 1-10, 12-15 (14) | |
32 | 0.00863796 | 41790 | (-5/8, -2/5) | [43, 210, -5670] | 1-9, 11-13, 16 (13) | |
33 | 0.00868573 | 2970 | (-4/7, -2) | [-21, -990, 10890] | 1-8, 10, 12, 14, 15 (12) | |
34 | 0.00870798 | 3930 | (1/5, 2) | [-59, 240, -12960] | 1-9, 11-13 (12) | |
35 | 0.00873414 | 1302 | (-1/11, 2) | [-31, 72, -2016] | 1-8, 10-12 (11) | |
36 | 0.00890612 | 281190 | (-1/3, 1) | [181, 2730, 436800] | 1-12, 15, 21 (14) | |
37 | 0.00923684 | 975 | (1, 1/2) | [50, 195, 8775] | 1-4, 6-9, 12, 16 (10) | |
38 | 0.00925154 | 27930 | (-3/5, -4/9) | [77, 1260, -220500] | 1-12, 16 (13) | |
39 | 0.00926186 | 430 | (-3/2, -1/3) | [-3, -80, -400] | 1-5, 7-11 (10) | |
40 | 0.00941651 | 7350 | (1/2, -3) | [-5, -120, -840] | 1-10, 12, 17 (12) | |
41 | 0.00942676 | 2550 | (-1/3, -1/2) | [-25, -120, 120] | 1-10, 12 (11) | |
42 | 0.00948236 | 431970 | (1, 2/7) | [341, 9240, 2827440] | 1-6, 8-10, 12, 20 (11) | |
43 | 0.00950035 | 168630 | (-3/5, -6/7) | [-83, -6720, -147840] | 1-12, 14 (13) | |
44 | 0.00950586 | 3476880330 | (11/59, -7/22) | [-15193, -948689280, -13637408400000] | 1-9, 12, 13, 15-18 (15) | |
45 | 0.00956986 | 1110 | (-5, 2) | [-11, 18, -90] | 1-8, 10, 11, 16 (11) | |
46 | 0.00962388 | 1020 | (-4/7, -3/2) | [-2, 30, -150] | 1-5, 7-11, 15 (11) | |
47 | 0.00962440 | 441210 | (-13, 4/11) | [541, 23100, 11226600] | 1-7, 9-11, 15, 17 (12) | |
48 | 0.00979228 | 104790 | (-1/3, -6) | [29, -1680, -84000] | 1-12, 16 (13) | |
49 | 0.00980065 | 17800530 | (-4/7, -6/11) | [359, -6270, -2664750] | 1-12, 14, 15 (14) | |
50 | 0.00983895 | 4998 | (-1/7, 2/3) | [-77, 504, -34272] | 1-6, 8-12 (11) | |
51 | 0.00985527 | 154770 | (-2/3, -2/5) | [-73, 330, -20790] | 1-10, 12, 15 (12) | |
52 | 0.00987802 | 19950 | (2/5, 1/2) | [535, 23940, 11371500] | 1-8, 10, 12, 15-16 (12) | |
53 | 0.00988886 | 861630 | (-2/3, -3/2) | [-53, 4620, 32340] | 1-12, 15 (13) | |
54 | 0.00991900 | 2090 | (1, 1/3) | [49, 190, 8360] | 1-4, 6, 8, 9, 12, 18 (9) | |
back to top or table