curve #4555

y² + xy = x³ − 108425663471230142228234132029275292635x + 382514153212130876922351378526573094321579755441733212497

a-invariants: [1, 0, 0, -108425663471230142228234132029275292635, 382514153212130876922351378526573094321579755441733212497]
c-invariants: [5204431846619046826955238337405214046481, -330492228375281077668718238816887723734277766207765316667129]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 6211004779178492809579744073240143386701604826069736785648084946384192134182454991888528130
discriminant (Δ): 18369620875029340952186658212655856173124724701381002479037718372866607001214779196654426879538805176468531200000000
Faltings height: 20.8455
naive height: 274.3510
primes of bad reduction: 2, 3, 5, 7, 17, 23, 73, 2417, 1311198157, 326962519007931381802849821884118161643340185389096681227585234200358259
regulator: 34076960793385080500624961.23655607136906692914958441742911338441009300288398063079194916
submitted by: apkorte
submitted at: 2026-10-18 01:39:43 UTC
last updated: 2026-10-18 01:39:43 UTC

Witness: 25 independent points log in to add more points →

(44071919552094710334, -284967329597884765789206087567)
(2521653136624377534, 11186454384769056643664238033)
(55215996290680872894, 403408673047076679040532250993)
(2427527429885599134, -11559110336677235165035013967)
(29974829595388842174, 155127593526369240902094264273)
(8006549126167913694, -5259012647342399528686337487)
(109007192752707183409896/26569, 11233143432023341050077540296504101/4330747)
(3844567260202343334, 4742339361932571740576035233)
(-9464869492072632066, -23682279501001741386897565167)
(-11666228457334988766, 7723255776483604214379540633)
(16411086036194275484, 54982055340708119104965094883)
(50228176978180415474286/25, -11256809101220000540174744887767699/125)
(1578536453781283614, -14672888341426990802393028687)
(3431901726699489024, -7129424706511190814613930377)
(855101830515805415454, -25003147624036850010335938400847)
(3987118299633816342, -3686715849378627969921965655)
(16221571959368836734, 53779309322173297807017819633)
(-101318157631300192131/16, 1827291640006154194697091496137/64)
(13766014115793426174, 38712115711615382769323585553)
(-11170767109254393342306/961, 253167115364885269917667918374063/29791)
(73741059151715990814, -627193812548198302154999057487)
(9115372317315751734, -12311387479389856545899885367)
(29249034028220242568256/2209, 3697400417286742449898153276631559/103823)
(-1659701904803791266, -23619834910867021166785244367)
(11265205289130010734, 24304032280352705666264517633)

Commentary

Specialization at T = -173/66 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -129423262682594003679387T^8 + 364444960853480119379208T^7 + 145088936906457036995136T^6 - 946405361864597551050816T^5 + 1086632441482958356509720T^4 - 615454597853371985855184T^3 - 2089672621773109442565264T^2 + 1503858169634937349756032T - 268043453574349038086592, A6 = 22378678518203765279639595291703734T^12 - 97642937907003790975195419097186584T^11 - 57701568157490976356217548246274864T^10 + 692728062356311676289756774010506240T^9 - 165059119295086271477298303488050440T^8 - 2065127205357525454766777049159397680T^7 + 866901377553305534044230525750471600T^6 + 2900595200522009433908965902519647040T^5 - 1511852767062133534934112666639840240T^4 - 1121738997234616988726901721424515200T^3 + 1300199883149489416130125996732379520T^2 - 448873241175573293059242892011302400T + 53141649442568035635166156713288960. 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 2026-10-18 01:39:43 UTC · history