curve #2999

y² + xy + y = x³ + x² − 195x + 2712

a-invariants: [1, 1, 1, -195, 2712]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/4ℤ
conductor (N): 39585
discriminant (Δ): -2896315695
Faltings height: 0.5015
naive height: 29.3932
primes of bad reduction: 3, 5, 7, 13, 29
regulator: 8.50939749504274205324257552843424877967839
submitted by: David Renshaw
submitted at: 2026-09-26 20:32:42 UTC
last updated: 2026-09-26 20:32:42 UTC

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

(6, 39)
(-10, 66)

Commentary

Cremona 39585h1 = LMFDB 39585.g4: https://www.lmfdb.org/EllipticCurve/Q/39585/g/4 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 39585, minimal discriminant -2896315695, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:32:42 UTC · history