curve #3380

y² + xy + y = x³ − x² − 282971359436387x − 189953825626075883101

a-invariants: [1, -1, 1, -282971359436387, -189953825626075883101]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 11232328230
discriminant (Δ): 1434543971556124814588210417337986527296000000
Faltings height: 7.3545
naive height: 111.4427
primes of bad reduction: 2, 3, 5, 17, 19, 67, 73, 79
regulator: 140.38933961339788042562465605188991682148680
submitted by: David Renshaw
submitted at: 2026-09-26 21:17:44 UTC
last updated: 2026-09-26 21:17:44 UTC

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

(17292219, 9349614802)
(1176557307, 40352354309746)
(61189873467/3481, 3357199906792334/205379)

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 = -183/85; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 11232328230, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:17:44 UTC · history