curve #2820

y² + xy + y = x³ − 19721x + 1064280

a-invariants: [1, 0, 1, -19721, 1064280]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/3ℤ
conductor (N): 474734
discriminant (Δ): -912438748
Faltings height: 1.0074
naive height: 41.2819
primes of bad reduction: 2, 13, 19, 31
regulator: 12.966359339984050889460103345345901338633648
submitted by: David Renshaw
submitted at: 2026-09-26 20:23:55 UTC
last updated: 2026-09-26 20:23:55 UTC

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

(49, 440)
(85, 20)
(36, 615)

Commentary

Cremona 474734e1 = LMFDB 474734.e1: https://www.lmfdb.org/EllipticCurve/Q/474734/e/1 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 474734, minimal discriminant -912438748, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:23:55 UTC · history