curve #543

y² + xy = x³ − 748560290571810810168611650732795790938065347391338016710x + 8271380284201252621251763373115565776243138253113288642252823365042992603044972259172

a-invariants

[1, 0, 0, -748560290571810810168611650732795790938065347391338016710, 8271380284201252621251763373115565776243138253113288642252823365042992603044972259172]

rank (lower bound)

≥ 29

torsion subgroup

trivial

conductor (N)

974577964543442160892570415381135671939306278462543535324631397980273407284075830036683707470545054553959558881958229183929792217811973208902830

naive height

404.5830

Faltings height

31.6510

discriminant (Δ)

-2710786471462749374614993166292988263188121797565799307100057421878289674658661348034237800500607508194038487108944963636095898561841141607117519789597580369102388428800000

primes of bad reduction

2, 3, 5, 13, 19, 23, 31, 127, 241, 147454289, 20554377348023, 1886568027092676434153747674819284839876889875232293, 55475379361618690930805426825153019584420146406248287036257

regulator

855753994549180338636678654638509471.993532750149501355044133541558785078092351885135831943387798

submitted by

Mundoamundo

submitted at

2026-09-02 23:15:14 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 19:08:56 UTC

Witness: 29 independent points log in to add more points →

(12764923916343910128018502444, -892201621382245053856737383957989945960982)
(544024912411844929335175670764, -400764692039090584129521311748663221770247702)
(20318429794111432649703945388, -1204177091010487294365456382976784591889814)
(16036849246742379039129168088, 625457593569114636133824303892456860231466)
(87791195160313344693665000524, -24883466897594838297601597582021025549118902)
(29253966742316885336780461804, 3377633540354457744230452977000902919871978)
(718932165594491252382810298636/49, -229297733994110398675579724462370189372783866/343)
(54222670865503667641072187164, -11273970468846699264482117348660956310428502)
(-30531611468201370216070739156, 1632542821925833904280282199075711433373418)
(-31640343181896987418340764116, 529697222543986246862906603827280455425258)
(21317907654223796752151274404236/1369, 31665715577546034225428530091659245868892371714/50653)
(-5862095345745666379209173876, 3529598572978841822032080308738291481411178)
(246275548529407628813307991564/25, -170168370078468078679308012324155199626821822/125)
(10841796261909760707886476124, -1195841187407231779902224629172525315138822)
(1228817851199913240439824392716/49, -787451786761089211095937738262123115718291706/343)
(117699018018179798292291569656/9, -22887506376673667805917674188581772883000534/27)
(14686865856400334005318528684, 667376838907779567436158764386428594174058)
(835534371623331993177259720636/49, -233924450667146660775487050705828197493844586/343)
(16850620560590856531494022124, -665054568332500746262530494302487655140822)
(-1180319481326411831956487066, -3025438184992674044313404078396960639976242)
(23652656497044410362611162245596/1369, -35660761716886260947025285884337878475319937726/50653)
(273134481541166735107720131484, 142057431667765465821019752417304659742719658)
(-1319951899884441331785093598652/49, -1022616100459927576764849811779798741279119426/343)
(13552949976283104399400090924, -784614051421290097428129231806739237606422)
(-30208060082372538649437813776, 1821611176443968203029651251697875084076298)
(157172699302652604246864013744204/10201, -647974542535012363626899961956561768879670739582/1030301)
(-28487278521932224930973843156, 2545129858954371526289281904377605796509418)
(14478482766835875781720999556733904/707281, 734799629824145619209353923916710538822891739044918/594823321)
(31432761881961468017459908204, 3974696689768925914049154262092987973821418)

Commentary

Fiber t = 18909/3310 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406): the 2-neighbor fibration that also carries wgxli's #391 and #390. Found by a Mestre-Nagao sieve over t = a/b and an iterated 'bootstrap' 2-covering point search (fake 2-descent over random cosets of the growing known lattice); 17 points are specializations of the generic sections, 12 were found on the coverings; the basis shown is LLL-reduced. Search run by Claude (Anthropic) with Seth Lupo, 2026-09-03.

last edited by Mundoamundo at 2026-09-02 23:15:14 UTC · history