curve #3326

y² + xy + y = x³ − 189181x + 31655304

a-invariants: [1, 0, 1, -189181, 31655304]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/6ℤ
conductor (N): 84058
discriminant (Δ): 24556367888
Faltings height: 1.4945
naive height: 48.0650
primes of bad reduction: 2, 13, 53, 61
regulator: 15.8652895402759803982128151088805353141043
submitted by: David Renshaw
submitted at: 2026-09-26 20:58:14 UTC
last updated: 2026-09-26 20:58:14 UTC

Witness: 2 independent points log in to add more points →

(459, 6218)
(-386, 7063)

Commentary

Cremona 84058j1 = LMFDB 84058.a2: https://www.lmfdb.org/EllipticCurve/Q/84058/a/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 84058, minimal discriminant 24556367888, torsion ℤ/6ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:58:14 UTC · history