curve #54

y² + xy = x³ − x² − 79x + 289

a-invariants: [1, -1, 0, -79, 289]
c-invariants: [3801, -232605]
rank (lower bound): ≥ 4
torsion subgroup: trivial
conductor (N): 234446 ★ record for rank ≥ 4
discriminant (Δ): 468892 ★ record for rank ≥ 4
Faltings height: -0.0870
naive height: 24.7291
primes of bad reduction: 2, 117223
regulator: 1.504344888275283974095271252282834672518585514
submitted by: David Renshaw
submitted at: 2026-05-28 21:07:17 UTC
last updated: 2026-07-01 23:03:09 UTC

Witness: 4 independent points log in to add more points →

(6, -1)
(4, 3)
(5, -2)
(8, 7)

Commentary

Smallest conductor of an elliptic curve of rank 4: conductor 234446 = 2·117223, Cremona label 234446a1 / LMFDB 234446.a1, y^2 + x·y = x^3 − x^2 − 79x + 289. By completeness of Cremona's tables below conductor 500000, this is the provably minimal conductor in rank 4. Generators [6,−1], [4,3], [5,−2], [8,7].

last edited by David Renshaw at 2026-05-28 21:13:43 UTC · history