curve #3366

y² + xy + y = x³ − 597217463667403x + 5603641587044871717506

a-invariants: [1, 0, 1, -597217463667403, 5603641587044871717506]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 27919716630
discriminant (Δ): 67416222327442166556471745892540167313062500
Faltings height: 7.2813
naive height: 113.6835
primes of bad reduction: 2, 3, 5, 13, 19, 103, 157, 233
regulator: 139.59816769622436289238891703411391667532589
submitted by: David Renshaw
submitted at: 2026-09-26 21:13:22 UTC
last updated: 2026-09-26 21:13:22 UTC

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

(368900734/25, 251162339917/125)
(919744345/49, 10880844905246/343)
(1316704285/49, 32460742331996/343)

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 = -5/103; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 27919716630, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:13:22 UTC · history