curve #2683

y² + xy = x³ − x² − 7848x + 222700

a-invariants: [1, -1, 0, -7848, 222700]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 349398
discriminant (Δ): 9888395954724
Faltings height: 1.2118
naive height: 38.5177
primes of bad reduction: 2, 3, 7, 47, 59
regulator: 5.3161952407993944300987564764092639418228401
submitted by: David Renshaw
submitted at: 2026-09-26 20:17:34 UTC
last updated: 2026-09-26 20:17:34 UTC

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

(26, 176)
(89, 428)
(21, 248)

Commentary

Cremona 349398c2 = LMFDB 349398.c3: https://www.lmfdb.org/EllipticCurve/Q/349398/c/3 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 349398, minimal discriminant 9888395954724, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:17:34 UTC · history