curve #41

y² + y = x³ − 79x + 342

a-invariants: [0, 0, 1, -79, 342]
rank (lower bound): ≥ 5
torsion subgroup: trivial
conductor (N): 19047851 ★ record for rank ≥ 5
naive height: 25.1942
Faltings height: 0.1081
discriminant (Δ): -19047851 ★ record for rank ≥ 5
primes of bad reduction: 19047851
regulator: 14.7905275701311284815007921692282023510586664287
submitted by: David Renshaw
submitted at: 2026-05-28 04:35:25 UTC
last updated: 2026-07-01 23:01:14 UTC

Witness: 5 independent points

(5, 8)
(4, 9)
(3, 11)
(7, 11)
(0, 18)

Commentary

Found by Brumer - McGuinness. A rank 5 example from their elliptic-curve examples page, with independent points of x-coordinates 5, 4, 3, 7, 0. This curve has rank exactly 5 (verified by both mwrank and the gp function ellrank().

last edited by John Cremona at 2026-07-03 12:26:42 UTC · history