Michael Stoll
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth, Germany

A New Record

On December 27, 2008, in the course of a systematic search for curves with many rational points in several families constructed by Noam Elkies, I discovered the following curve of genus 2.

y2 = 82342800 x6 - 470135160 x5 + 52485681 x4 + 2396040466 x3 + 567207969 x2 - 985905640 x + 247747600

It has at least 642 rational points, with x-coordinates in the following list.

0, -1, 1/3, 4, -4, -3/5, -5/3, 5, 6, 2/7, 7/4, 1/8, -9/5, 7/10, 5/11, 11/5, -5/12, 11/12, 5/12, 13/10, 14/9, -15/2, -3/16, 16/15, 11/18, -19/12, 19/5, -19/11, -18/19, 20/3, -20/21, 24/7, -7/24, -17/28, 15/32, 5/32, 33/8, -23/33, -35/12, -35/18, 12/35, -37/14, 38/11, 40/17, -17/40, 34/41, 5/41, 41/16, 43/9, -47/4, -47/54, -9/55, -55/4, 21/55, -11/57, -59/15, 59/9, 61/27, -61/37, 62/21, 63/2, 65/18, -1/67, -60/67, 71/44, 71/3, -73/41, 3/74, -58/81, -41/81, 29/83, 19/83, 36/83, 11/84, 65/84, -86/45, -84/89, 5/89, -91/27, 92/21, 99/37, 100/19, -40/101, -32/101, -104/45, -13/105, 50/111, -113/57, 115/98, -115/44, 116/15, 123/34, 124/63, 125/36, 131/5, -64/133, 135/133, 35/136, -139/88, -145/7, 101/147, 149/12, -149/80, 75/157, -161/102, 97/171, 173/132, -65/173, -189/83, 190/63, 196/103, -195/196, -193/198, 201/28, 210/101, 227/81, 131/240, -259/3, 265/24, 193/267, 19/270, -279/281, 283/33, -229/298, -310/309, 174/335, 31/337, 400/129, -198/401, 384/401, 409/20, -422/199, -424/33, 434/43, -415/446, 106/453, 465/316, -25/489, 490/157, 500/317, -501/317, -404/513, -491/516, 137/581, 597/139, -612/359, 617/335, -620/383, -232/623, 653/129, 663/4, 583/695, 707/353, -772/447, 835/597, -680/843, 853/48, 860/697, 515/869, -733/921, -1049/33, -263/1059, -1060/439, 1075/21, -1111/30, 329/1123, -193/1231, 1336/1033, 321/1340, 1077/1348, -1355/389, 1400/11, -1432/359, -1505/909, 1541/180, -1340/1639, -1651/731, -1705/1761, -1757/1788, -1456/1893, -235/1983, -1990/2103, -2125/84, -2343/635, -2355/779, 2631/1393, -2639/2631, 396/2657, 2691/1301, 2707/948, -164/2777, -2831/508, 2988/43, 3124/395, -3137/3145, -3374/303, 3505/1148, 3589/907, 3131/3655, 3679/384, 535/3698, 3725/1583, 3940/939, 1442/3981, 865/4023, 2601/4124, -2778/4135, 1096/4153, 4365/557, -4552/2061, -197/4620, 4857/1871, 1337/5116, 5245/2133, 1007/5534, 1616/5553, 5965/2646, 6085/1563, 6101/1858, -5266/6303, -4565/6429, 6535/1377, -6613/6636, 6354/6697, -6908/2715, -3335/7211, 7363/3644, -4271/7399, -2872/8193, 2483/8301, -8671/3096, -6975/8941, 9107/6924, -9343/1951, -9589/3212, 10400/373, -8829/10420, 10511/2205, 1129/10836, 675/11932, 8045/12057, 12945/4627, -13680/8543, 14336/243, -100/14949, -15175/8919, 1745/15367, 16610/16683, 17287/16983, 2129/18279, -19138/1865, 19710/4649, -18799/20047, -20148/1141, -20873/9580, 21949/6896, 21985/6999, 235/25197, 16070/26739, 22991/28031, -33555/19603, -37091/14317, -2470/39207, 40645/6896, 46055/19518, -46925/11181, -9455/47584, 55904/8007, 39946/56827, -44323/57516, 15920/59083, 62569/39635, 73132/13509, 82315/67051, -82975/34943, 95393/22735, 14355/98437, 15121/102391, 130190/93793, -141665/55186, 39628/153245, 30145/169333, -140047/169734, 61203/171017, 148451/182305, 86648/195399, -199301/54169, 11795/225434, -84639/266663, 283567/143436, -291415/171792, -314333/195860, 289902/322289, 405523/327188, -342731/523857, 24960/630287, -665281/83977, -688283/82436, 199504/771597, 233305/795263, -799843/183558, -867313/1008993, 1142044/157607, 1399240/322953, -1418023/463891, 1584712/90191, 726821/2137953, 2224780/807321, -2849969/629081, -3198658/3291555, 675911/3302518, -5666740/2779443, 1526015/5872096, 13402625/4101272, 12027943/13799424, -71658936/86391295, 148596731/35675865, 58018579/158830656, 208346440/37486601, -1455780835/761431834, -3898675687/2462651894


Michael Stoll,  Jan 9, 2009