curve #3978

y² + xy = x³ − x² − 725220385853626099441226087775x + 135888679495399615706574796105000776535601661

a-invariants: [1, -1, 0, -725220385853626099441226087775, 135888679495399615706574796105000776535601661]
c-invariants: [34810578520974052773178852213209, -117407819084025111322877279451483191621924875677]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3482515236760862602092865033854708270
discriminant (Δ): 16434051353085032541995456749719241595533322651472652326633471134668206310957507214945610000
Faltings height: 16.2014
naive height: 217.8824
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 31, 59, 79, 113, 173, 181, 251, 1231, 330217, 25236637
regulator: 11561124087184194.3861251305268400805384068382268884038654720
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:38:22 UTC
last updated: 2026-10-05 04:38:22 UTC

Witness: 11 independent points log in to add more points →

(89873310175354327/529, 51018257136786333462928777/12167)
(-6197322072811246217/26569, -74049240317197522319968026259/4330747)
(14897047826750538063/97969, 165497535324163769236297104666/30664297)
(16548189298873303665873/17106496, -1303818138892365649928306719965939/70752467456)
(1357524819688509290442/724201, 45133352758707627700948913319003/616295051)
(105413529171940436390673697/12434926144, 1076938589557417636113853132452889117293/1386643484169728)
(17406862694787002130432751/14593848025, -54845282759193812738382601596982055614/1763009810660125)
(8288510874105526830313/43877376, -689876887793675163675713842782331/290643738624)
(-1485313019314538765246753/3812939001, -4462808033732284190206910369104403725/235445170372749)
(1199491400022817693503169797898/1585065652771689, -285321626776037376709208072775253730626479589/63106033381117879869387)
(-12948602965307639228368015729782/21087260013549961, -57261777285116000074253104363876936305587037129/3062176572942711231683291)

Commentary

Unlisted rational-isogeny neighbor of curve#3598 (source contributor: dujella). Source equation and witness provenance are credited to that entry. Exact rank 11: 11 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:38:22 UTC · history