curve #3693

y² + xy = x³ − 350157604483734638786832426845429476320x + 2510435188740399342636197095094554753287459379850011238400

a-invariants: [1, 0, 0, -350157604483734638786832426845429476320, 2510435188740399342636197095094554753287459379850011238400]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 120600636120040262281305547153244997893031818350820655996701181065730
discriminant (Δ): 25121477374297765527923282858800055355877054995479383055579862153334243664419119090461139063381908579398890000000000
Faltings height: 20.9879
naive height: 277.8679
primes of bad reduction: 2, 3, 5, 13, 37, 61, 127, 173, 313, 337, 491, 541, 1721, 2161, 2423, 3469, 3907, 4547, 8221, 13477, 203713, 1365868604929
regulator: 71386248750004333047936.2426575487895370085139415317925700501538279329591
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:44:30 UTC
last updated: 2026-09-30 01:44:30 UTC

Witness: 17 independent points log in to add more points →

(800217818841917979390, 22630524388919943871694782174230)
(5298998356394073012914940/579121, 3753734203522087893404393107446081780/440711081)
(6202442093718931977556920/543169, 390019107850172647869363372610633320/400315553)
(211302933108280610926636110/4012009, 2899244872652760548492957636732977080210/8036054027)
(265170418237568144511583360/1830609, 4283665989768466336499891229482661088960/2476813977)
(1688533734929422327635758720/173106649, 10917887793990137573275726797924721195840/2277564180893)
(412617494424846529298617367160/6004855081, 256092319882702733363319661507175578920437880/465322225081771)
(935028922007813837873439755640/432738993241, 11955828145014979117789028732272083966169530720/284668259184733789)
(13965179968744727832820167894840/556395154561, 40523587438277921090670039596075449578551287320/415025717294986559)
(29102909738762127678599902429560/2499981563689, 13561080137382122919935700613206364498147226760/3952803349740279637)
(410906959932442342750047359448240/8909660881, 8329428611173434662084585048157672899816882824080/840991800218471)
(271102523916019909110833715373103835/26890470860192164, 9821210036248151877367420189920826550691720110006565/4409583901513753776499288)
(29106171926731708797638610635850806810/668012582313014689, 144423775393265094724004170220604855799851342263303171990/545980288936690874489870863)
(326229636668925011902265778590246334040/45080374253149152009, 5706504470178611986985163297714687414160012026083803647320/302678288302741133985059184027)
(29036189118070977912280984113326997748890660/1653128578637135139540529, 89646478154072703642657221390080867576998130283054224929880893840/2125494330094195199532633703278826583)
(68456839435017863108945575192537856070462346740/4535836359563298452915354281, 7868675074789570992759096597765727747219907118031214033841824250607920/305482310730302600094306871579307726017429)
(2558503412531632913047226631053005866385803390927429862060598426921803129849/105191608143388803730163181285001030737602266595726361600, -98776205186262709269303402410496914878420052374432464504622278214963919966269111380719127036811251203257625372413/1078876272418027519324878504896246485994821511401013635043010581567410634808078336000)

Commentary

Specialization at t = -942/47, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:44:30 UTC · history