curve #2961

y² = x³ + x² − 644x − 4704

a-invariants: [0, 1, 0, -644, -4704]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 2184
discriminant (Δ): 8414025984
Faltings height: 0.6111
naive height: 31.0183
primes of bad reduction: 2, 3, 7, 13
regulator: 1.700598484763839251509225858109905838132
submitted by: David Renshaw
submitted at: 2026-09-26 20:30:26 UTC
last updated: 2026-09-26 20:30:26 UTC

Witness: 1 independent point log in to add more points →

(31, 78)

Commentary

Cremona 2184m2 = LMFDB 2184.j4: https://www.lmfdb.org/EllipticCurve/Q/2184/j/4 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 2184, minimal discriminant 8414025984, torsion ℤ/2ℤ × ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:30:26 UTC · history