curve #3483

y² = x³ + x² − 2474617489821035784x + 1498329225569769638818813956

a-invariants: [0, 1, 0, -2474617489821035784, 1498329225569769638818813956]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 62271688747599408
discriminant (Δ): 13308324430688211366524792624427609457403470008075264 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 6
Faltings height: 9.1741
naive height: 138.6715
primes of bad reduction: 2, 3, 7, 11, 19, 23, 31, 37, 47, 73, 97, 101
regulator: 258595.49169646766155061057770627460179125788453536
submitted by: jesper-petersen
submitted at: 2026-09-27 09:21:03 UTC
last updated: 2026-09-27 09:21:03 UTC

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

(674259543, 11676071296890)
(913558680, 259588068894)
(95325319890/121, 8177722022535984/1331)
(1457985521106/1225, 662879333844199296/42875)
(3199776381576/3481, 116170441649063802/205379)
(75801564176893344/42941809, 14432395998653961301607634/281397674377)

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) = (1971,52217). PARI/GP 2.17.4 gives exact rank 6 and torsion Z/2Z x Z/4Z. Conductor N = 62271688747599408.

last edited by jesper-petersen at 2026-09-27 09:21:03 UTC · history