curve #5160

y² + xy = x³ − 2375353961891265636493611632951308935x + 1409075027759502737646659471099166373605792729134616201

a-invariants: [1, 0, 0, -2375353961891265636493611632951308935, 1409075027759502737646659471099166373605792729134616201]
c-invariants: [114016990170780750551693358381662828881, -1217440823984210365497739268285850872622944955544802640985]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 70406956101573874140909238243890165565935985526040058829451404712463857460420580090848878
discriminant (Δ): 25666437320706702824811827158129450688897621471866484725947693239325375691710673274553454137482130568937472
Faltings height: 19.5468
naive height: 262.8882
primes of bad reduction: 2, 3, 13, 17, 29, 31, 101, 1217, 374056178400297913150562417197, 1284587135853784193084942879863116960492221369803
regulator: 8149153888758334152716316.7936302695067032452537956428860279172631484132062178832655461
submitted by: apkorte
submitted at: 2026-10-21 05:42:25 UTC
last updated: 2026-10-21 05:42:25 UTC

Witness: 24 independent points log in to add more points →

(893291760926288760, -3329079650760372970366131)
(937167829672612590, 77902441859785068857891229)
(-825104908043225514, 1675488400463382347854254693)
(1062930126458559798, 291821757438510433884351909)
(771221475893282268, 189367702833152183942317269)
(15340060567984746630, 59789255409384487472555213061)
(881203890951813686, -13288403289300791037129307)
(4355906110942931022, -8585503002031073872544230731)
(932521919786250258, -70169630473315442210999811)
(44474949543320436, -1141717676397233783334708057)
(22081279477756912494/25, 1211397810347507862410634393/125)
(5898936215672390124, 13880379766651891672499446965)
(868915188353857926, -33714699969156138989742219)
(1489771344091707721110/1681, 249615690790022979627479717949/68921)
(958947651312080466, -114300392307364978656790659)
(43938010496030230134/49, -3519379366581476705538986637/343)
(698956218981419126070/2209, -86189389063905396622328233515093/103823)
(4076753134300730166, 7712369182884180511917606693)
(137609029866519624, 1041542448318052263580184715)
(876387347401339104438/961, 1072963422613776533141109437499/29791)
(588714946904945910, -463366578766262718923234331)
(1241175729301001166, 610652019607407146863776693)
(45758697342939631077366/57121, -1945158841376213023445447339349141/13651919)
(1122264835184879526, -395938227617432220383065707)

Commentary

Specialization at T = -1172/133 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -55167841848013120672432T^8 - 332957956141194498177568T^7 + 30491322260931582308864T^6 + 1529425586966988026755040T^5 - 1952037697734560475124240T^4 + 2598926506840431770721968T^3 - 1867830095998069273733704T^2 - 277424385594374102335720T - 14777266661889408077275, A6 = 4598281008482828350515082915932544T^12 + 42978899384640031706121578829901184T^11 + 92492192628018200121094722387755072T^10 - 190123782916092378560803318012553920T^9 - 804112508567836069639299650095483056T^8 + 453677299928650241856349297890606048T^7 + 3341413915638233675031424423024721088T^6 - 4626422467332761770277917962033627744T^5 + 1280672988092416347275859838092852864T^4 + 274908311165313855430793313344325200T^3 + 204841690501474358540727164780007400T^2 + 26424146202979804837762201204291000T + 968816204643834101037115185253750. 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 2026-10-21 05:42:25 UTC · history