curve #3255

y² + y = x³ + x² − 153x + 678

a-invariants: [0, 1, 1, -153, 678]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 7733
discriminant (Δ): 935693
Faltings height: 0.0259
naive height: 26.7114
primes of bad reduction: 11, 19, 37
regulator: 3.56084541369148350141051218875886944893509
submitted by: David Renshaw
submitted at: 2026-09-26 20:46:43 UTC
last updated: 2026-09-26 20:46:43 UTC

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

(6, 3)
(74, 632)

Commentary

Cremona 7733d1 = LMFDB 7733.c2: https://www.lmfdb.org/EllipticCurve/Q/7733/c/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 7733, minimal discriminant 935693, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:46:43 UTC · history