curve #3389

y² + xy = x³ − 1141810422400x + 424861718219592032

a-invariants: [1, 0, 0, -1141810422400, 424861718219592032]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/10ℤ
conductor (N): 168077130
discriminant (Δ): 17292010058959530010410479652819300000
Faltings height: 5.8801
naive height: 94.9045
primes of bad reduction: 2, 3, 5, 13, 17, 101, 251
regulator: 57.4703368547680757432023940354103330263062
submitted by: David Renshaw
submitted at: 2026-09-26 21:19:14 UTC
last updated: 2026-09-26 21:19:14 UTC

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

(-17939611/16, 34828792247/64)
(325034369/256, 4138027701143/4096)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/10ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with d=t^2/(t-(t-1)^2), c=td-t, b=cd, at t = 13/30; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 168077130, torsion ℤ/10ℤ.

last edited by David Renshaw at 2026-09-26 21:19:14 UTC · history