curve #3484

y² = x³ − x² − 9543695128779220260x + 11343572956219572510053641092

a-invariants: [0, -1, 0, -9543695128779220260, 11343572956219572510053641092]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 36629127870174840 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 6
discriminant (Δ): 44325377496302257936580620114275673108602447562500000000
Faltings height: 9.6343
naive height: 142.7208
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 61, 433, 1153
regulator: 354936.65584004754744812871785952434446904672021624
submitted by: jesper-petersen
submitted at: 2026-09-27 09:23:45 UTC
last updated: 2026-09-27 09:23:45 UTC

Witness: 6 independent points log in to add more points →

(539615864, 79691730186250)
(1918157964, 9736232779650)
(-901848981031/361, 960122536333220260/6859)
(28121603625474/6241, 120595012183693047000/493039)
(175491232086164/10201, 2289626869595479618750/1030301)
(41939467959614584/23126481, 59142452740669590256250/111215247129)

Commentary

Found by a conductor-oriented search in the standard Z/2Z x Z/4Z family y^2 = x(x+u^2)(x+v^2), with (u,v) = (7625,73346). PARI/GP 2.17.4 gives exact rank 6 and torsion Z/2Z x Z/4Z. Conductor N = 36629127870174840.

last edited by jesper-petersen at 2026-09-27 09:23:45 UTC · history