curve #3376

y² + xy = x³ − 1684883951828x + 841748501204713872

a-invariants: [1, 0, 0, -1684883951828, 841748501204713872]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/10ℤ
conductor (N): 404284650
discriminant (Δ): 29151794524080184451026155017856000
Faltings height: 5.6789
naive height: 96.0718
primes of bad reduction: 2, 3, 5, 7, 11, 17, 29, 71
regulator: 102.185202789146415934879054482652377369825
submitted by: David Renshaw
submitted at: 2026-09-26 21:17:02 UTC
last updated: 2026-09-26 21:17:02 UTC

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

(753718, 872200)
(30195041632756/40896025, 3909575829194178836/261530079875)

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 = 14/11; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 404284650, torsion ℤ/10ℤ.

last edited by David Renshaw at 2026-09-26 21:17:02 UTC · history