curve #5978

y² = x³ + x² − 9581151308834651913195280244505025x + 358559054589281911030747372371379682354940812431619

a-invariants: [0, 1, 0, -9581151308834651913195280244505025, 358559054589281911030747372371379682354940812431619]
c-invariants: [459895262824063291833373451736241216, -309795023165139573889937306673251796554909572358366080]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 8695817217975270702029545916305160192036786620186268139060193767787379278516836054936696
discriminant (Δ): 750330787098362628719549912478826355733233216408508001633786954797313945286282955279611630172051148032
Faltings height: 18.3770
naive height: 246.3489
primes of bad reduction: 2, 3, 7, 11, 13, 19, 4297, 4433494036989996753503482173009288536566211848419182927816620424488210611555703
regulator: 8781930743394985947625472.7583067436271862537678471742624121300392099296528429242424972
submitted by: apkorte
submitted at: 2026-10-27 14:51:29 UTC
last updated: 2026-10-27 14:51:29 UTC

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

(51879414612319043, -1061419486809136817191830)
(51387341553492239, -1380434418907215642519822)
(52677348210729950, 154735841653322592922287)
(99388176831748127, -19699321517808771211381134)
(96063831472556081, -18018294486471877166690814)
(60723399427049927, -816250130433002031682734)
(178694600410338041, 65973420270204672315307014)
(104347325649227055431/1681, 119417476071069018829195312194/68921)
(-35205664336150405, 25538883787598566754130762)
(-7046394180992599, -20633025909298367382822186)
(95176235664076043, 17573177788398908974550070)
(747410914376080390/9, -314427912434661591189497951/27)
(41925178978669307, 5528139221498752826559306)
(-165367927691230267/4, -209227577366361463273266903/8)
(1570021289682156137, -1963508691212624255716293354)
(487364643496384319, 333841262238552151260704142)
(112805617320696161, 26706084529571955607187214)
(60306248852337107, 283121350912736675314794)
(2399849300941141001/64, -3692541828762973677838998651/512)
(72167101894572665, -6554993251331805328957062)
(2464116183377350121, -3865040631592981113128888214)
(62228162686923059, 1819472183301659731419498)
(-23291223786847423, 23855413327605854443359534)
(52692052111885793, -74180511366558213744270)

Commentary

Specialization at T = 113/64 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -299179532755056721600T^8 + 17595572277256973050240T^7 - 417198241841028433831120T^6 + 1640729978369826838604080T^5 - 3100722994060599052732840T^4 + 3447605855872936235820160T^3 + 2242797006715822622440160T^2 - 6780974649680701579123160T - 3861433387934633535827467, A6 = 4059456937701100750406882080000T^12 - 282164040891383692833079679552000T^11 + 6335982179671782376102889433278336T^10 - 57358833600342341658935952553773440T^9 + 596470207133623029963843326957746320T^8 - 2651283194034110828378464332085379520T^7 + 3864693449523759808880763001015666800T^6 + 2106280696838105770175445350531933136T^5 - 6473880048439824158203228245177202440T^4 - 4211251865396149188236650636780633280T^3 + 4195370517911125633418237460366830320T^2 + 7286118774738610034794421910368398280T + 2836757717663966250804688091791547974. It is a 2-neighbour of the fibration given in curve#4187 (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 2026-10-27 14:51:29 UTC · history