curve #154

y² = x³ − 3926559x + 3073474946

a-invariants: [0, 0, 0, -3926559, 3073474946]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 716225953128355512
naive height: 57.2153
Faltings height: 2.6751
discriminant (Δ): -206273074500966387456
primes of bad reduction: 2, 3, 1663, 5669, 5801, 60631
regulator: 61186.124504384347188884237562748022432111958022609102918
submitted by: Edgar Costa
submitted at: 2026-06-24 21:09:10 UTC
last updated: 2026-07-01 22:54:24 UTC

Witness: 9 independent points

(-26, -56352)
(67, -53016)
(-202, -62116)
(263, -45376)
(265, -45294)
(397, -39714)
(509, -34738)
(-551, -71202)
(-563, -71454)

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 21:09:10 UTC · history