curve #5979

y² + xy + y = x³ − x² − 53277273977331919052247265607945627x + 5071306541591271749034694912633046049357517000319179

a-invariants: [1, -1, 1, -53277273977331919052247265607945627, 5071306541591271749034694912633046049357517000319179]
c-invariants: [2557309150911932114507868749181390081, -4381608851934858779658085225411257271359485316959515521]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 175363906724941939917345509202231003059281386917758326349323466305344130
discriminant (Δ): -1431787493201184053584244540330170275725781799953750242511415787925919593827345043463445607727380480000000
Faltings height: 18.9222
naive height: 251.6340
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 29, 83, 107, 131, 197, 1704979069, 2794911869439479530045299668088511856914879
regulator: 65513038297862832884.88498139782941564370784031589718164847190093171257498182581112
submitted by: apkorte
submitted at: 2026-10-27 15:48:03 UTC
last updated: 2026-10-27 15:48:03 UTC

Witness: 22 independent points log in to add more points →

(206897969748634187, -53897871145559874719641094)
(-217180536364445813, 79989095773272193240286906)
(3070382155266954187, -5365327520510661044099841094)
(483520293044461787, -303897645411620862472225094)
(146819192887658027, 20346901539037755388414906)
(1114606684822080187, -1153437279056312400107169094)
(172575527524074187, -31884968410479180760041094)
(196310888065251787, -46667115524483382099426694)
(76077215022001307, 38189465199978505273742506)
(-267953836896061943, -10405270946601873859719994)
(-57937741836924563, -89238932742834149718115594)
(7544003790973387, 68336022188231471502590906)
(164770414487977867, -27680095963128032706971974)
(1278322877293253779/25, -6225922445845527086554031918/125)
(124797132275787787, 19133387877792112546277306)
(189661375787021707, 42297254894091423301145786)
(116064153531926987, 21241751687520230741246906)
(-134669826896840693, -99014043912700181006751814)
(143506532282986987, 19520787314947780182166106)
(-242730165861345503, -60845281862773129311295354)
(200092548274146187, 49214143793423376372158906)
(3203839701838889779/25, 2332993447036899749018130898/125)

Commentary

Specialization at T = -337/2 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -11871310334515632T^8 + 4710110008611306816T^7 - 2086020069231986711136T^6 - 184523251173755635536480T^5 - 6701046568565117730395760T^4 - 153188863150147099034594256T^3 - 1728397061227994797149412104T^2 + 1293824158270290221749735140T - 3009001448051781638619102975, A6 = 425207191402126274866560T^12 - 563052339092973212554731264T^11 + 242123500866097746692387835264T^10 + 23032729293888429602372703530880T^9 + 3121622944690084624163225355490944T^8 + 295262495155831884321522320323442496T^7 + 13000284347332506536750445881902721952T^6 + 266703534992086688942874014282698665168T^5 + 2231811615310432924685764525429401885924T^4 + 6596027232624269118843692038876188450360T^3 + 95451184990094031890361698435840315288700T^2 - 68941255711464655644860348072488527433500T + 38303805195121251570249945560744186021250. It is a 2-neighbour of the fibration given in curve#4187 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-27 15:48:03 UTC · history