curve #4584

y² + xy = x³ − 3154654875492219172169918480800576457378665x + 2211349431963903802705549767404432101994598275908377750849174617

a-invariants: [1, 0, 0, -3154654875492219172169918480800576457378665, 2211349431963903802705549767404432101994598275908377750849174617]
c-invariants: [151423434023626520264156087078427669954175921, -1910605909216812885537822134188464775903729144515456018238618132969]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 263542669493980362273701944576786465311279689102958423623867133463127042831994298942466417531070
discriminant (Δ): -103251438448619135013708103418624349100934288891270198726609406040931178740946698796726672519734789357661214436874275450880000000
Faltings height: 23.3419
naive height: 305.2361
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 41, 47, 59, 73, 6131, 9817, 231634667493284287, 73315279431040153541, 1274688633367886253095517054987462043
regulator: 261972076220345130819397420.766729406083792756321921866610388339048588932836350343097904654
submitted by: apkorte
submitted at: 2026-10-18 10:10:32 UTC
last updated: 2026-10-18 10:10:32 UTC

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

(1829019158853637047674, 50597067988048777442533772524163)
(1658146586779736493674, -39235959552271556492107735237837)
(835854426750147150074, -12589244492208765694856835293437)
(2420737565799412568294, 93595984200228088036035847015883)
(-895902514794901316326, -65715479870968262259914200315837)
(-1773871829736722354326, 47176265446489071447267341656163)
(1134918730977452197934, -9638145212538340983214813104457)
(555135975611633739532234/441, -141954764817780437875358834089242457/9261)
(712362053763253668074, -18044058217217321562944660705437)
(3821042942433669880634, 214349983086881688770459088376643)
(12858510271847233235306/25, 3365379054206591863181065507315831/125)
(1225242437973827481794, -13619459160942709604113376188117)
(41233000900971474464906/49, -4237163868736538695309166711345131/343)
(-212314035600410283526, 53586905179504664664807726728963)
(931059472892719916474, 9015945521812805316267199928963)
(2816761882405392222458, -125196014634594877881200897431549)
(781485972384986151674, 14943234793846288355670364732163)
(1388296256612008991954, -22527973036501932480569221364557)
(1042556488068650487674, -7458097899114135446135934931837)
(2115123536620868619002, 70720530596725732707888433928963)
(725027369142069576284714/961, 481055651534937670477070183451599933/29791)
(1104558750773214654074, 8629294576971800994935764175363)
(-1925392752930963656326, 33876302427778987150323848812163)
(-1758048004863212424326, 48205057115199807427775473836163)
(7548163430357341287158, 639104360076950716860359427508511)
(1284036151347569945174, -16664682393342092949235099374337)

Commentary

Specialization at T = 1634/921 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -94521237307405223348592T^8 + 194603007768925915807776T^7 + 374678018163025368194124T^6 - 569211691896707346596088T^5 + 55463370301014723924120T^4 - 1135612908646444410668928T^3 - 2702507543729482509654336T^2 + 609092474805433024526016T - 184694377125002807554992, A6 = 11265741534554236438342976060251776T^12 - 34833166213267258548363680994916992T^11 - 69502033189823387959197312852122064T^10 + 265587490260133017697161120524802480T^9 + 273144084685427886803017231992231060T^8 - 1114854493674664679669934784078225824T^7 - 766273999597311933358466543970856272T^6 + 2407762999977059159006184076291662816T^5 + 1357060199643920217443950358505612000T^4 - 540509467373796107556447739286515200T^3 + 808090270114106954000628418781159424T^2 - 154315127645781466989761857773289728T + 30433563297067405810172392059479424. 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 10:10:32 UTC · history