curve #1926

y² + xy = x³ − 1812692499920238359044828488x + 37598312198452856879141331946397639732592

a-invariants: [1, 0, 0, -1812692499920238359044828488, 37598312198452856879141331946397639732592]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 7957055249285451655668179172815993925972577312050
discriminant (Δ): -229489933413739800898518566909955415893954796036949450416087357552789560265374750000
Faltings height: 14.7067
naive height: 200.3787
primes of bad reduction: 2, 3, 5, 17, 19, 23, 53, 59, 71, 73, 79, 107, 331, 601, 3319, 5813, 214519, 3616013, 17506729
regulator: 72280614626567.07439906823765824681784663891467356962167390293613
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:16:23 UTC
last updated: 2026-09-25 19:16:23 UTC

Witness: 13 independent points log in to add more points →

(8366842780548126, 765308965663908409349580)
(511158033364057022202/16410601, -7084445285945617642913527026876/66479344651)
(59828716021279119/4, 14633990375802694038322797/8)
(487254941474136, 10716209233243029139884)
(64539305014470783/4, 16395886938695041518347217/8)
(25162104088242, -88983578989818354846)
(27538112356414728/169, -4431617318142953658310332/2197)
(148236534100952, 1739612251020893159324)
(554622147565827/4, 12526989911562438622551/8)
(15118052585219013318/100489, 56716277040164128019345692572/31855013)
(28395796033752, 94982083859833906524)
(-38230250287674, -225881586997012113906)
(12772558913736, 128566254751927993284)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (4,-1,145/6).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:16:23 UTC · history