curve #3993

y² + xy = x³ − x² − 2583167927740578324x + 14625458243583407363235377664

a-invariants: [1, -1, 0, -2583167927740578324, 14625458243583407363235377664]
c-invariants: [123992060531547759561, -12636395921898099689443401383685]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 2521187658201578085610417610020422
discriminant (Δ): -91303382004233281769658736489447812533563219971277027953648
Faltings height: 9.9935
naive height: 143.2283
primes of bad reduction: 2, 3, 17, 31, 61, 73, 109, 139, 151, 307, 349, 499, 1237, 1637, 3301
regulator: 1219567982656528980.4487046649563902644378045864663287657490531
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:41:27 UTC
last updated: 2026-10-05 04:41:27 UTC

Witness: 12 independent points log in to add more points →

(-8924235840440203270204208/3494881525655529, 13969766734839534592286167126364157400/206608738975215825734667)
(-1809480509764373252735/650782637521, 9274897564468370931599644243668/524993512297203431)
(5944988358231585145/3576398809, 26128757286663440197225287369/213879377974627)
(1359675005543365840/3208429449, 21199079193183436666658533016/181735069279707)
(5303412166341514615044025/736328684932129, 12148009865482978402976878435239923139/19980556461547539827633)
(-19511509993062829757900/8147194040241, 1956222263364099732583166661847412/23254772217687053289)
(152372785354712807199097/62542205339769, 74672165000498429000939958283358420/494606462647431588147)
(4187962152359250806478865/1300478446249609, 9344664683496033010885474573221400321/46898044898984998450523)
(1312539149876888503936305/51148715259889, 1501419811035693775375909374470628219/365807069703277726537)
(4241556759317619307010177521/1113413157785763025, 287944784545346061076189195104252183122859/1174855688247288893554633625)
(-3049266202452143892455247180/1858739367325758241, 304604892905630262590293194605991621008148/2534123322039366433681894961)
(2655767856513108516675367613652/758420835212741453809, 4600575925841793153854333446364331398009584296/20886486183526129055522610621623)

Commentary

Unlisted rational-isogeny neighbor of curve#3630 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 12: 12 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 04:41:27 UTC · history