curve #3398

y² + xy = x³ − 1378024147874891x + 19689438735404357150577

a-invariants: [1, 0, 0, -1378024147874891, 19689438735404357150577]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/7ℤ
conductor (N): 61672312002
discriminant (Δ): 12886301283878849892317524094184479932416
Faltings height: 7.1781
naive height: 116.1919
primes of bad reduction: 2, 3, 7, 19, 43, 1797293
regulator: 175.71232655300406086655367641603665116586857
submitted by: David Renshaw
submitted at: 2026-09-26 21:20:43 UTC
last updated: 2026-09-26 21:20:43 UTC

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

(20127214, 10347628129)
(-1062012254/25, 4953390239141/125)
(403028721694/18769, 812046681978713/2571353)

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 = 133/4; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 61672312002, torsion ℤ/7ℤ.

last edited by David Renshaw at 2026-09-26 21:20:43 UTC · history