curve #153

y² + xy = x³ − x² − 1240549x + 521297989

a-invariants: [1, -1, 0, -1240549, 521297989]
c-invariants: [59546361, -450133503885]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 616136865466771394
discriminant (Δ): 4929094923734171152
Faltings height: 2.3725
naive height: 53.7068
primes of bad reduction: 2, 83, 14549, 255114981391
regulator: 102053.46086070287983607086017344691046650256242692048241
submitted by: Edgar Costa
submitted at: 2026-06-24 21:09:07 UTC
last updated: 2026-07-01 22:52:29 UTC

Witness: 9 independent points log in to add more points →

(-10, -23097)
(62, 21055)
(73, 20727)
(183, 17240)
(195, 16837)
(210, 16327)
(278, 13927)
(-286, -29057)
(-410, -30793)

Commentary

Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).

last edited by Edgar Costa at 2026-06-24 21:09:07 UTC · history