curve #2875

y² = x³ − 21x − 20

a-invariants: [0, 0, 0, -21, -20]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 288
discriminant (Δ): 419904
Faltings height: -0.2235
naive height: 20.7472
primes of bad reduction: 2, 3
regulator: 1.240339345699340752600006946273422038963
submitted by: David Renshaw
submitted at: 2026-09-26 20:26:22 UTC
last updated: 2026-09-26 20:26:22 UTC

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

(8, 18)

Commentary

Cremona 288b1 = LMFDB 288.b3: https://www.lmfdb.org/EllipticCurve/Q/288/b/3 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 288, minimal discriminant 419904, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:26:22 UTC · history