Michael Stoll
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth, Germany
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.
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
This paper can be cited as a reference for this curve.
Update 2026-09-04: This record has now been broken. Open AI's GPT-6 Astra has found a curve with at least 648 points; see the announcement on Epoch AI's website. This curve is highly symmetric (its rational automorphism group has order 12); in this regard it is similar to the curves of Keller and Kulesz, and Stahlke that held records prior to December 2008. The curve given here has minimal automorphism group of order 2.