curve #2951

y² + y = x³ + x² − 63x + 138

a-invariants: [0, 1, 1, -63, 138]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 6293
discriminant (Δ): 5292413
Faltings height: 0.0079
naive height: 24.0588
primes of bad reduction: 7, 29, 31
regulator: 4.48197772581653863960624379418198564956107
submitted by: David Renshaw
submitted at: 2026-09-26 20:29:47 UTC
last updated: 2026-09-26 20:29:47 UTC

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

(6, 3)
(-4, 18)

Commentary

Cremona 6293f1 = LMFDB 6293.d2: https://www.lmfdb.org/EllipticCurve/Q/6293/d/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 6293, minimal discriminant 5292413, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:29:47 UTC · history