curve #1193

y² + xy + y = x³ − x² − 2450825x + 1477391753

a-invariants: [1, -1, 1, -2450825, 1477391753]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/6ℤ
conductor (N): 2327418 ☆ record for ℤ/6ℤ torsion, rank ≥ 3
discriminant (Δ): 4170909418426368
Faltings height: 2.2344 ☆ record for ℤ/6ℤ torsion, rank ≥ 3
naive height: 55.7494
primes of bad reduction: 2, 3, 31, 43, 97
regulator: 29.278919428549781670610254101077596423864261
submitted by: David Renshaw
submitted at: 2026-09-23 21:43:28 UTC
last updated: 2026-09-23 21:43:28 UTC

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

(-1797, 9628)
(-1409, 46876)
(-89, 41212)

Commentary

Found by a rational-parameter search of the Kubert-Tate family for torsion 6, at t=4/93. 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 21:43:28 UTC · history