curve #4277

y² + xy + y = x³ + x² − 20845656078219069716013684908066351890x + 31463060765353331858086739177380841883984296285897372831

a-invariants: [1, 1, 1, -20845656078219069716013684908066351890, 31463060765353331858086739177380841883984296285897372831]
c-invariants: [1000591491754515346368656875587184890721, -27184084501265278732891378837415912485527358557919216806545]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 139883182700156075678240143523566742427129552398645127416507706029578508671711831046708338596405138
discriminant (Δ): 152083955373867280930311974970586224458386900726854737681312823930051742678695328601122965967784699064123047411712
Faltings height: 20.4415
naive height: 269.4042
primes of bad reduction: 2, 3, 7, 11, 17, 19, 29, 41, 2417, 118643471271063007, 13154440207536530510113316253120360112142374009561177739898271696127
regulator: 675552251745816771672819641.4338960459713444037902137550538777541836534417901974597844364
submitted by: apkorte
submitted at: 2026-10-16 08:23:05 UTC
last updated: 2026-10-16 08:23:05 UTC

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

(-2963791074259093519, 8198238768336341577216049491)
(1469350597212947669, -2001449109157791658982545581)
(-4244925192654891175, 6592446384431972403826510707)
(3464623825039022575, -910343805161394513098631033)
(4379865367338308249, 4917501126084878417666861811)
(25782887969646502889/25, 415736007743752361407461509127/125)
(108754883494123334641/25, -600837297398942189550744496729/125)
(-92360059976517777511/25, 952397045510555029754082092103/125)
(-4933096136084511175, -3774603761866059386331253293)
(6699315156700765399, 13873796255383843242489260289)
(-1888478018493618689, -8005913559040357870981568583)
(1262586989250574361, 2675131630899847277278910067)
(6300948488000652001/9, 112016032032709163852235082273/27)
(1740030122671711097, -677708540978997416707735821)
(33961756486786070129/25, -297169834305664678578962514393/125)
(15184105502609755249/9, -28267297132113862139791884191/27)
(40225633551772314201, -253539139274396117421618481613)
(16050285443116016153, 61900121574771199559138998899)
(3571481766423556121, -1602857321075756450593053069)
(109043022844621460129/25, -606219884544809375101292784393/125)
(7924928136995196113, 19078358741535993866217080667)
(9401268147942768481/9, 88845189504244750043317772833/27)
(201125852238465521233/121, -1578205936563334418497148618391/1331)
(6310199776447645711, 12295782247407015168247600617)
(-53172261685998452111/25, 1016874440240253144435268249127/125)

Commentary

Specialization at T = -1790/379 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -48393752778478434415347T^8 + 234108952651394609784372T^7 - 80814256417092311126712T^6 - 679388493916744317276048T^5 + 849130008583102656447600T^4 - 1612877972465689819445088T^3 - 1721372370991450068420576T^2 + 3195581161738120242427584T - 1022262851511189811019952, A6 = 4288640503583697356313758651467086T^12 - 31081375829532514009215543683973804T^11 + 28276823001661393553469125282330460T^10 + 256926134808356645699750469379877640T^9 - 449617863421297320058465556298389964T^8 - 1048634128416718070202997112516438256T^7 + 2065736410495836489928705601739523680T^6 + 2236546605261081463863864681224077632T^5 - 3826369071427309561101765341099280384T^4 - 533037124876780800676744763435971200T^3 + 3060828262346510699070310919653538688T^2 - 1792851678299100321486790706293201152T + 349240517064527681543842990887408000. 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 2026-10-16 08:23:05 UTC · history