### Nontorsion points of low height on elliptic curves over Q

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 (a1, a2, a3) (where E is Y2 + a1 XY + a3 Y = X3 + a2 X2 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 (a1, a2, a3) = (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) (a1, a2, a3) 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)

We birationally parametrize pairs (E,P) by P1*P1 as follows. Let A,u be affine coordinates on the two P1's. Then P is the point (0,0) on the curve
E: Y2 + a1 XY + a3 Y = X3 + a2 X2
whose coefficients ai are given by
a1 = c(Au5 + 2A2u4 + 3Au4 - 2u4 - 4Au3 - 8u3 - A2u2 - 10Au2 - 10u2 - 4A2u - 10Au - 6u - A2 - 2A - 1),
a2 = c2 u (u + 1) (u + A + 1) (Au - A - 1) (Au2 + u2 + u + A + 1) (Au3 - Au2 - 2u2 - Au - 2u - A - 1),
a3 = c (u + 1)2 (u + A + 1) (Au2 - Au - 2u - A - 1) a2.
Here c is any nonzero scalar; different choices of c yield isomorphic (E,P), and in our arithmetic setting we always choose the smallest c that makes each ai an integer.
One of several nice things about this parametrization is that it lets us automatically exclude torsion points over Q. Indeed if some A,u yield a nonsingular elliptic curve E then P cannot be a torsion point of order 8 or less, and P is a torsion point of order 9, 10, or 12 if and only if A equals -1, -1/2, or 0 respectively. By a celebrated theorem of Mazur, this covers all possible orders of a torsion point on an elliptic curve over Q.