curve #6004

y² + xy = x³ − 35376670386207552738280424549044535x + 2555012810129013423461855950892650379207999062397097

a-invariants: [1, 0, 0, -35376670386207552738280424549044535, 2555012810129013423461855950892650379207999062397097]
c-invariants: [1698080178537962531437460378354137681, -2207531067951467600418163809378193724791901757442298329]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 2104871013643958885399236871022985406085174767280449508928964787218548041758610
discriminant (Δ): 13414653947308735616180691060034653324080080129684632659072712124687133451985010829712315689268684800000
Faltings height: 18.6616
naive height: 250.2677
primes of bad reduction: 2, 3, 5, 7, 17, 23, 47, 59, 83, 163, 241, 2663, 9712731331, 109618608364842654238983000805193366395233621421111
regulator: 9702062839560899066813.73854858693154339205545026864797267758263377346080912758388916
submitted by: apkorte
submitted at: 2026-10-27 18:38:16 UTC
last updated: 2026-10-27 18:38:16 UTC

Witness: 23 independent points log in to add more points →

(84677136692450284, -12906244331369113417967867)
(54688706200715014, -27997757879643237270070427)
(125489260168447414, -9579643556677052874814187)
(355201412813009134, -186558936896700927360335867)
(140892605970528214, -19170866096226584887397387)
(118770725757807334, -5360736024273885720308267)
(5571684846760627036/49, 552061136274680199752613139/343)
(-87954876531657026, 70612599890037595542060253)
(57129273578768644, -26840738072732119419853337)
(252067791815064334, 98252838270947843307412933)
(60421410218951254, -25260399221986886747670347)
(82814331924133974, 13902368877468279481643253)
(833726931416726566/9, 230380803555039107400081271/27)
(151482568288632694, -25925463323971455674450507)
(119067657434106214, 5552518709928895749990613)
(318526972287806374, -153636437280230296542840827)
(-75886824816932426, -69300924768219816859359947)
(-1336766796955946, 51012751149364322354589493)
(137703098604351862, 17166291820106300748413365)
(118615373654384034, 5259991322869431893518833)
(114956931622484134, 2717762531147970237801733)
(62237239272659284, 24379106511517786798326133)
(15948406373632165414/121, 17955232852005940607457660623/1331)

Commentary

Specialization at T = -181/119 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -16161338919951574537347T^8 - 183967925781776423843232T^7 - 306908503380230554874889T^6 + 1321820887548439573004394T^5 - 1089630203923766659668705T^4 + 1528127126557525645958556T^3 - 1325190033898795951823799T^2 - 1776684907545050991600198T - 509996388400821572993532, A6 = 794074859864541402077984985828990T^12 + 13480216201562664173283363261525120T^11 + 58813044515050204101637768724085915T^10 - 27005959952131974781942342196787600T^9 - 297164658623174446064912117521429860T^8 - 610242665718369417817232061166604040T^7 + 2737840349733935589023201371120885290T^6 - 1476924630602948597826251203809007800T^5 - 1760224581565992882116105923868957910T^4 + 421401656201140614180762210583159560T^3 + 1429992029749290001354466937891106371T^2 + 752996037739894365105522529181137224T + 139570007340666087279594607715201844. It is a 2-neighbour of the fibration given in curve#4191 (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 18:38:16 UTC · history