curve #3381

y² + xy = x³ − 454805738544185x + 3677688141636234951225

a-invariants: [1, 0, 0, -454805738544185, 3677688141636234951225]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 265642379310
discriminant (Δ): 177881007094520248546025436567964487961000000
Faltings height: 7.2883
naive height: 112.8663
primes of bad reduction: 2, 3, 5, 7, 11, 31, 97, 167, 229
regulator: 132.64996868989984666901851282524895882227495
submitted by: David Renshaw
submitted at: 2026-09-26 21:17:54 UTC
last updated: 2026-09-26 21:17:54 UTC

Witness: 3 independent points log in to add more points →

(37527670, 198629374315)
(177030445/16, 13905513835/64)
(4421258422/841, 922860057828023/24389)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/6ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with c=(10-2t)/(t^2-9), b=c+c^2, at t = 68/33; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 265642379310, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:17:54 UTC · history