curve #53

y² + y = x³ − 7x + 6

a-invariants: [0, 0, 1, -7, 6]
c-invariants: [336, -5400]
rank (lower bound): ≥ 3
torsion subgroup: trivial
conductor (N): 5077 ★ record for rank ≥ 3
discriminant (Δ): 5077 ★ record for rank ≥ 3
Faltings height: -0.5614 ★ record for rank ≥ 3
naive height: 17.4513
primes of bad reduction: 5077
regulator: 0.41714355875838396981711954461809339674981011
submitted by: David Renshaw
submitted at: 2026-05-28 21:07:16 UTC
last updated: 2026-07-01 23:04:07 UTC

Witness: 3 independent points log in to add more points →

(1, 0)
(2, 0)
(0, 2)

Commentary

Smallest conductor of an elliptic curve of rank 3: conductor 5077 (prime), Cremona label 5077a1 / LMFDB 5077.a1, y^2 + y = x^3 − 7x + 6. Cremona's tables are complete below conductor 500000, so 5077 is the provably minimal conductor in rank 3. Generators [1,0], [2,0], [0,2].

last edited by David Renshaw at 2026-05-28 21:13:24 UTC · history