curve #1191

y² + xy + y = x³ − x² − 832936109x + 213650223049109

a-invariants: [1, -1, 1, -832936109, 213650223049109]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/9ℤ
conductor (N): 14049882 ☆ record for ℤ/9ℤ torsion, rank ≥ 2
discriminant (Δ): -19682229955776531123284168933376 ☆ record for ℤ/9ℤ torsion, rank ≥ 2
Faltings height: 4.6840 ☆ record for ℤ/9ℤ torsion, rank ≥ 2
naive height: 79.5139 ☆ record for ℤ/9ℤ torsion, rank ≥ 2
primes of bad reduction: 2, 3, 7, 11, 31, 109
regulator: 14.4344489739583389452553415984744983031214
submitted by: David Renshaw
submitted at: 2026-09-23 21:40:14 UTC
last updated: 2026-09-23 21:40:14 UTC

Witness: 2 independent points log in to add more points →

(-3159, 14707018)
(-87283337/1369, 141101983096/50653)

Commentary

Found by a rational-parameter search of the Kubert-Tate family for torsion 9, at t=-4/7. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t^2*(t-1);b=c*(t^2-t+1); Converted to the global minimal model; PARI/GP ellrank returned bounds [2,2]. The 2 supplied points and the torsion subgroup are independently certified by the site.

last edited by David Renshaw at 2026-09-23 21:40:14 UTC · history