curve #3370

y² + xy = x³ − 16602286815530x + 38543763023078004900

a-invariants: [1, 0, 0, -16602286815530, 38543763023078004900]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/7ℤ
conductor (N): 24896975610
discriminant (Δ): -348912656234165824852274138454589440000000
Faltings height: 6.6716
naive height: 103.7198
primes of bad reduction: 2, 3, 5, 13, 79, 808081
regulator: 182.32108503701838749132267507262071579342474
submitted by: David Renshaw
submitted at: 2026-09-26 21:16:09 UTC
last updated: 2026-09-26 21:16:09 UTC

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

(-1404325260/289, 10410720921510/4913)
(580132540/49, 13289137029610/343)
(25900409860/3721, 3660739552339870/226981)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/7ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=t^3-t^2, c=t^2-t, at t = 79/40; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 24896975610, torsion ℤ/7ℤ.

last edited by David Renshaw at 2026-09-26 21:16:09 UTC · history