curve #3040

y² + xy + y = x³ − 3169x + 211076

a-invariants: [1, 0, 1, -3169, 211076]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/6ℤ
conductor (N): 20130
discriminant (Δ): -17259290492400
Faltings height: 1.2244
naive height: 38.0456
primes of bad reduction: 2, 3, 5, 11, 61
regulator: 2.38440484978054775239540980049951444880656
submitted by: David Renshaw
submitted at: 2026-09-26 20:34:29 UTC
last updated: 2026-09-26 20:34:29 UTC

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

(31, 362)
(70, 542)

Commentary

Cremona 20130g1 = LMFDB 20130.e3: https://www.lmfdb.org/EllipticCurve/Q/20130/e/3 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 20130, minimal discriminant -17259290492400, torsion ℤ/6ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:34:29 UTC · history