curve #4226

y² + y = x³ − 417389016912604118381708877469167x + 3255206773058249324786174571271655071684264490666

a-invariants: [0, 0, 1, -417389016912604118381708877469167, 3255206773058249324786174571271655071684264490666]
c-invariants: [20034672811804997682322026118520016, -2812498651922327416615254829578709981935204519935640]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 80151163480449143652638260373196573000359294842516282152443251449303489021117
discriminant (Δ): 76117407996287559815040621810294938793138761104990662832222746415859210066543089838851172111389557
Faltings height: 17.6024
naive height: 236.9483
primes of bad reduction: 3, 7, 11, 13, 23, 29, 43, 67, 79, 265523065253, 220716503477760798348504972643581656125444870309526437
regulator: 8435367235421159453016.17527464202440828415537502339993852026987618707270564646249323
submitted by: apkorte
submitted at: 2026-10-14 14:43:19 UTC
last updated: 2026-10-14 14:43:19 UTC

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

(4761433121225049, 1172938405070423032553011)
(14559052061632347, -514239565302132303549416)
(-8679564207241659, 2494812219150031981015315)
(12729485296188237, -68866674523597078867085)
(9214828759715694, -437604669733570053776177)
(8674139363463843, -536063927632546069913657)
(633166680814137066/49, 47746537282684040027933749/343)
(133278390080710143, 48115148773166224839621031)
(1174810852313367, 1663272400248361775732359)
(10870222716010927, -50394949342484263632616)
(5720447451626484, -1027009253153685793556219)
(3659499446439105, -1332958994051177626580948)
(1498718202688783233/16, 1794309941478750410786973727/64)
(382196718583963552, -235950971046584771085618010)
(12734287525269579, -71228917130274794894504)
(-14262521085450369, -2511364451779602322258856)
(-6860131556901972456/1369, 115738884549605188247327477725/50653)
(10701044018773377, -118786170244522889724116)
(-20877239955770487/4, 18402624487508550668600945/8)
(14829318989190744, -571583925976820286041474)
(13662222786173230, 320757572536491038976527)
(37202379203097223, 6262273407609015047334968)
(10206135094154268, 241665299981935808572093)

Commentary

Specialization at T = -485/27 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -13129724922135998400T^8 - 393315956007325349760T^7 + 5388075614914218900720T^6 + 133624136603638823580000T^5 - 2980942426742659655805540T^4 + 33575588298751816593482040T^3 - 178476287434090182905197255T^2 - 27064681539446356514316710T - 38737534397522382035015527, A6 = 17001575848734905085648825600T^12 + 808662010058021550277497323520T^11 + 2204274099561464081452163602176T^10 - 450617120852864904310146381112320T^9 - 6041711991147376183046964974244960T^8 + 125500678000542015284784371883062400T^7 + 2667323428245326791597400104572570240T^6 - 56072697432225196572060046394633385120T^5 + 291328099318551627699751437189468287265T^4 - 366248679959978292773278702918063389000T^3 + 804324836109783714230677491953667748735T^2 + 249729199613745513617677147674877497230T + 91023269067533367175532101155650107690. 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 2026-10-14 14:43:19 UTC · history