curve #56

y² + y = x³ − 277x + 4566

a-invariants: [0, 0, 1, -277, 4566]
c-invariants: [13296, -3945240]
rank (lower bound): ≥ 6
torsion subgroup: trivial
conductor (N): 7647224363
discriminant (Δ): -7647224363
Faltings height: 0.5828 ★ record for rank ≥ 6
naive height: 30.3760 ★ record for rank ≥ 6
primes of bad reduction: 7647224363
regulator: 159.18394480133901410572598227360268100133182426854
submitted by: David Renshaw
submitted at: 2026-05-29 00:03:59 UTC
last updated: 2026-07-01 22:58:19 UTC

Witness: 6 independent points log in to add more points →

(-22, 3)
(-21, 33)
(-19, 54)
(-14, 75)
(-12, 78)
(-7, 78)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=6: [0,0,1,-277,4566], |Delta|=7647224363; their I-column records 49 integral x-coordinates. The submitted points certify rank >= 6 here.

last edited by David Renshaw at 2026-05-29 00:13:05 UTC · history