curve #2963

y² + xy + y = x³ + x² − 12789x + 551346

a-invariants: [1, 1, 1, -12789, 551346]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 609
discriminant (Δ): 1472026689
Faltings height: 0.9441
naive height: 39.9826
primes of bad reduction: 3, 7, 29
regulator: 3.513460239082121967725501233614427425684
submitted by: David Renshaw
submitted at: 2026-09-26 20:30:35 UTC
last updated: 2026-09-26 20:30:35 UTC

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

(-104, 929)

Commentary

Cremona 609b2 = LMFDB 609.a4: https://www.lmfdb.org/EllipticCurve/Q/609/a/4 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 609, minimal discriminant 1472026689, torsion ℤ/2ℤ × ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:30:36 UTC · history