curve #3624

y² + xy = x³ + x² − 1576373756450214922x + 759926798530798069794077800

a-invariants: [1, 1, 0, -1576373756450214922, 759926798530798069794077800]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 672519215577201888821790774558
discriminant (Δ): 1226727845993834425057233609652398817063367323001819772
Faltings height: 9.2505
naive height: 137.3186
primes of bad reduction: 2, 3, 11, 41, 53, 61, 71, 89, 103, 191, 499, 1013, 5653
regulator: 27477669180.0133225324285649571169600248936365346934505018796
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:21:42 UTC
last updated: 2026-09-29 19:21:42 UTC

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

(-1421837642, 11263734044692)
(-1036404363, 35783898016732)
(408741102754, 261318455289453262)
(-96556513372113/73441, 470921814710516251681/19902511)
(-46070557043/144, 60640894748153695/1728)
(145837229314/169, 14273443950074278/2197)
(941823615477/1369, 51833408920992100/50653)
(3191540073454/3721, 1414559400681173902/226981)
(9855074830156/11881, 6219678376863383776/1295029)
(12672084477346/3481, 42718516724929271936/205379)
(54838010696629/70225, 42145142761670670103/18609625)

Commentary

Specialization at T = -26/27 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:21:42 UTC · history