curve #4006

y² + xy = x³ − 4392076037989488171378581214035900x + 46671635543273329318653514757313287975016864406032

a-invariants: [1, 0, 0, -4392076037989488171378581214035900, 46671635543273329318653514757313287975016864406032]
c-invariants: [210819649823495432226171898273723201, -40324293109388156847546111485561829149672418257396449]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 216095858582728308893742177217643607485136038653170
discriminant (Δ): 4481374339012390278609292200373604626354568421949817199606571622276114685170561160216601825865463360000
Faltings height: 18.3908
naive height: 244.0089
primes of bad reduction: 2, 3, 5, 7, 29, 31, 37, 67, 79, 139, 157, 211, 241, 257, 509, 709, 2801, 12301, 16529, 50263, 53617
regulator: 57628563213127232063.194151972195750545818795269632274534739956704290
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:44:17 UTC
last updated: 2026-10-05 04:44:17 UTC

Witness: 15 independent points log in to add more points →

(-66693294685778516, 6553028237693922154776928)
(1837243956133593448/169, -1004446946908920825989789180/2197)
(4728043801450864, -5100137277981833025024692)
(47333700900215092048/10201, -5292965201746106204810891866444/1030301)
(-388198946675292026456/57121, -119176367177471066913678306902068/13651919)
(206559596378800276507/676, -2900532392118782705489403461557/17576)
(5545962403792712527024/11881, 408926574987127995382677133628692/1295029)
(4638847445058061786/9, 9909926135245940546202587696/27)
(2158682621705379592002691/3319684, 3155390786859440142168870830737059109/6048464248)
(-193015984748414647741514/8720209, 297020515159760173037565122487839296/25750777177)
(-252672162148169174792108/26224641, 1260487931641043180667556270626011188/134296386561)
(978718824189990343432/114921, 122482816347500229066140008802012/38958219)
(2801669911452742227693568/8532241, -4596138085025071715926257058444688164/24922675961)
(441378011219593127766080239/7283939716, 1082235383926119634349655899207889282653/621655119001736)
(57679337162506514938567664776/14389681849, 13850682743126322951420236376528205790861444/1726143065560493)

Commentary

Unlisted rational-isogeny neighbor of curve#3650 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 15: 15 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 04:44:17 UTC · history