curve #6159

y² = x³ + x² − 73883347220217960986993218180x + 7727333107659711288181154889630383989448864

a-invariants: [0, 1, 0, -73883347220217960986993218180, 7727333107659711288181154889630383989448864]
c-invariants: [3546400666570462127375674472656, -6676415805018011831392517247413416020930654400]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 3926046641084574152321158849702341670078001371260205141923329094823925574696
discriminant (Δ): 16436994288460819926329379071628171110063674101886421266334267681712584548658235939072
Faltings height: 15.3191
naive height: 211.0305
primes of bad reduction: 2, 3, 17, 23, 16937, 87211, 97021, 285308551219757980245712093, 601908179713562561098137159667
regulator: 7737484678066917461957.100202739566303733762608671743997076995874938603478541358878
submitted by: apkorte
submitted at: 2026-10-29 19:32:44 UTC
last updated: 2026-10-29 19:32:44 UTC

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

(143624319731756, -280289534426136311256)
(160530776191934, -60671402062327276902)
(638084652022949/4, 213301382364506511423/8)
(178625733643214, 478857566957446620522)
(4081263310649774/25, -16097473193154016305582/125)
(90591992953754, -1333256800529044815582)
(56474680831766, 1932592666061439035406)
(153926704112534, -42007226176994179698)
(68071572404572202, 17760092233624013201171346)
(1216665485006786, -41458839789702140202906)
(606582943156949/4, 821665815014470553577/8)
(139649471340506, -364696044377642017506)
(7228589094071606/49, -67209635374189499736894/343)
(1884781718287265/16, 51938805982592717876313/64)
(4461467803348022, 297459669298498961968494)
(152351256719276, 85574055430587702456)
(130495653070124, 555070270903347798888)
(19463359405861859/121, 92596412371775011596108/1331)
(162342751806044, 107134856348039872992)
(115553184159419, 856040539460793141492)
(132088604137372, -522290656036616008944)
(151905838045499, 96471858345707447388)

Commentary

Specialization at T = -47/26 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -2350252713284250372T^8 - 50405464050571118556T^7 - 952448555730502901313T^6 - 695049490520215525224T^5 - 2931749924460790132683T^4 - 2386411337210880522384T^3 + 2409436881372804430248T^2 + 4544024515422355802304T - 2684309556378273707952, A6 = 1416965056476304965893460864T^12 + 45023054173800546171881279808T^11 + 917608982150765091196202404680T^10 + 4283108867178449968605589545480T^9 + 53149163760305922465025229930778T^8 + 64588783908002903981500312561392T^7 - 30251100417436562854623526202130T^6 - 127076856645436090494117187739472T^5 + 56548211197412673092440132446600T^4 + 156704525825325938035598980257408T^3 - 11739117611564936890190411272416T^2 - 136493880790478548113078471588096T + 53534237645294085780370164274560. It is a 2-neighbour of the fibration given in curve#3589: 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 2026-10-29 19:32:44 UTC · history