curve #3396

y² + xy + y = x³ − 22400761384755293x + 1290453262980776137939556

a-invariants: [1, 0, 1, -22400761384755293, 1290453262980776137939556]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 464963829330
discriminant (Δ): 9094461543021742283391523416554033430562500
Faltings height: 7.8387
naive height: 124.5572
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 43, 59, 359
regulator: 110.49291936257169371379789012919448555392587
submitted by: David Renshaw
submitted at: 2026-09-26 21:20:24 UTC
last updated: 2026-09-26 21:20:24 UTC

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

(99122985, 209575228882)
(21965625225/256, 39880256329597/4096)
(45722877135/529, 3502852995584/12167)

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 = 157/177; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 464963829330, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:20:24 UTC · history