curve #80

y² + y = x³ − 402157x + 96291336

a-invariants: [0, 0, 1, -402157, 96291336]
c-invariants: [19303536, -83195714520]
rank (lower bound): ≥ 9
torsion subgroup: trivial
conductor (N): 157107745029925477
discriminant (Δ): 157107745029925477
Faltings height: 2.0879
naive height: 50.3274
primes of bad reduction: 157107745029925477
regulator: 1253079.4942741755784427259681451604931159440062484019892
submitted by: David Renshaw
submitted at: 2026-06-02 18:44:32 UTC
last updated: 2026-07-01 22:51:30 UTC

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

(-9, 9995)
(16, 9479)
(46, 8825)
(61, 8484)
(74, 8181)
(-76, 11243)
(102, 7505)
(108, 7356)
(-113, 11844)

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,-402157,96291336], |Delta|=157107745029925477, I=131. The submitted integral points give the rank >= 9 witness checked by the site.

last edited by David Renshaw at 2026-06-02 18:44:32 UTC · history