curve #3625

y² + xy = x³ + x² − 628194012441407122x + 175424217648985174555974640

a-invariants: [1, 1, 0, -628194012441407122, 175424217648985174555974640]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 686044836958354728450143389446
discriminant (Δ): 2571557733180790736761999668359933846719299683095776372
Faltings height: 9.1787
naive height: 134.5585
primes of bad reduction: 2, 3, 11, 17, 31, 101, 157, 199, 347, 487, 593, 2351
regulator: 141086050529.491999328583483130949484065661915660899888573554
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:21:53 UTC
last updated: 2026-09-29 19:21:53 UTC

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

(-1482167189/4, 151224869583521/8)
(321243547, 2602232340742)
(342655102, 634125949546)
(621800606, 5021969024810)
(634992070, 5706194944000)
(-1552757380843097/15342889, 927079705887704426814251/60098096213)
(-322025129766/1369, 892098157001095042/50653)
(8213282257/9, 513810198946984/27)
(8620679761/49, 2876881576926601/343)
(26937795382/81, 1322221417754768/729)
(21081992078679/62500, 21569076167632305967/15625000)

Commentary

Specialization at T = 38/9 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 19:21:53 UTC · history