curve #4201

y² + xy = x³ − 24678177866799656847971717213285435898453935785x + 1490865482566543141765428173320569297663667597141222159551847045163225

a-invariants: [1, 0, 0, -24678177866799656847971717213285435898453935785, 1490865482566543141765428173320569297663667597141222159551847045163225]
c-invariants: [1184552537606383528702642426237700923125788917681, -1288107776937493274485331718577778282756701857893655302404180535704402921]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 22304343890526450064144705503043731669398709375695492452300742909568725455497738810473799887632978110
discriminant (Δ): 1678633134172064369840906248841182960042419955994408007191716970466236153637086837214614706007412336065984601844110479915278055918080000000
Faltings height: 25.4419
naive height: 332.0803
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 53, 109, 173, 101879, 104094700537, 179223765898916789751764060783309600874223337494008677165715327850089829
regulator: 1457829711754532561521663348.36028392005093788127981553030549607463079261404615890058645543
submitted by: apkorte
submitted at: 2026-10-13 01:17:56 UTC
last updated: 2026-10-13 01:17:56 UTC

Witness: 26 independent points log in to add more points →

(39617401382202929769410, 23986688830862259751299215116589675)
(96614375654265489664570, -2903434761603873374590258328905085)
(33068784112791927263690, -26663651903091527599052146308461485)
(145529772814556147420420, 31330791880519508034122249283740915)
(88448694210615705358010, -249357535397227912286824449372925)
(77150382761814007890170, -6793209839528874049386911099409085)
(162209676525737298702170, 41903267910838047248107241257466915)
(-81033071976229982404870, -54392295634196179674693222684061885)
(165732392332066649108299010/1681, -276926021910903334843587503997827274125/68921)
(27486836098182050805050, 28867065594393181976963868929728835)
(-178442369259461460764230, -14580580987885853218891076564313085)
(87040550824690128262250, -1512449838534983307076038732212365)
(-5811740916310907333830, 40423908263820665342950052814830915)
(172610556771885536798420, -48723507440982878237567773136319085)
(87442235926075549538330, -1243548723319505904546274841832445)
(377148530211402206186543/4, 11998500651117331948619637727049233/8)
(17244885916868513519059370, 71609811609476054410162539034982497715)
(87897946183865775474170, -898980951169620013568758098609085)
(116228473055591614962170, -13881488721612623168987472614193085)
(-168875866053205829531230, -29021259463114854308381769393183085)
(34931871574494148673313530/289, -81309857290213722687099191804454263405/4913)
(837979193854080106775380355/6724, 10329131332982582749730068614860085431795/551368)
(-174071002408074764300830, 22630620910394871940839717766640915)
(2208255138066509836498346/25, -57161475697816069419775885448640521/125)
(-108046582327291443091330, 53813711516634163910898811143793415)
(84267799518260828805170, 3111367801738691391876527654165915)

Commentary

Specialization at T = -11584/4313 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -154503213541262535600T^8 + 4159635009352858651680T^7 - 21187086453270000393696T^6 - 687438318305734637863392T^5 - 4576470188855512350897360T^4 - 19322004287067422371639920T^3 - 39029689014386535692132184T^2 + 4186478995367154403195512T - 1679949255743694529867467, A6 = 747141710909816776767093360000T^12 - 29305404314392232038648503024000T^11 + 301598848438544320323778700280000T^10 + 2292679730804159433657382244285760T^9 - 4484131884783219751251582549192624T^8 + 544733850758018343321283421195576736T^7 + 7256453148985237089497289163544883552T^6 + 31881087967205708307154627785355135968T^5 + 53417637701365905114499013007709555776T^4 + 36362990355034819792007607408188452080T^3 + 65566855132366254216911881386291117144T^2 - 854195394262226327944905852404523432T + 424114741036883895390965869842474630. It is a 2-neighbour of the fibration given 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 2026-10-13 01:17:56 UTC · history