curve #81

y² + y = x³ − 826609x + 289956150

a-invariants: [0, 0, 1, -826609, 289956150]
c-invariants: [39677232, -250522113816]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 172539371946838571
discriminant (Δ): -172539371946838571
Faltings height: 2.1805
naive height: 52.4936
primes of bad reduction: 69238607, 2491953253
regulator: 91567.683284080923712615686709609971136528575064325022194
submitted by: David Renshaw
submitted at: 2026-06-02 18:44:33 UTC
last updated: 2026-07-01 22:52:28 UTC

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

(17, 16610)
(-57, 18354)
(60, 15510)
(103, 14349)
(-143, 20130)
(-188, 20945)
(243, 10170)
(248, 10010)
(331, 7253)

Commentary

Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=9: [0,0,1,-826609,289956150], |Delta|=172539371946838571, I=120. The submitted integral points give the rank >= 9 witness checked by the site.

last edited by David Renshaw at 2026-06-02 18:44:33 UTC · history