curve #1192

y² = x³ + x² − 783365x + 172533900

a-invariants: [0, 1, 0, -783365, 172533900]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/6ℤ
conductor (N): 2899380
discriminant (Δ): 17867432910467250000
Faltings height: 2.3966
naive height: 52.3277 ☆ record for ℤ/6ℤ torsion, rank ≥ 3
primes of bad reduction: 2, 3, 5, 11, 23, 191
regulator: 6.0653921506742468871880135820817622418024670
submitted by: David Renshaw
submitted at: 2026-09-23 21:43:26 UTC
last updated: 2026-09-23 21:43:26 UTC

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

(-935, 9405)
(-737, 18711)
(4609, 307395)

Commentary

Found by a rational-parameter search of the Kubert-Tate family for torsion 6, at t=-45/23. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t;b=t*(t+1); Converted to the global minimal model; PARI/GP ellrank returned bounds [3,3]. The 3 supplied points and the torsion subgroup are independently certified by the site.

last edited by David Renshaw at 2026-09-23 21:43:27 UTC · history