curve #2122

y² = x³ + x² − 30867739684x + 2071092196593056

a-invariants: [0, 1, 0, -30867739684, 2071092196593056]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 2428381032 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 4
discriminant (Δ): 29275254698157592852374082808064
Faltings height: 4.8606
naive height: 84.0725
primes of bad reduction: 2, 3, 7, 11, 19, 23, 31, 97
regulator: 54.75539704104194838318060822481907502756167071
submitted by: jesper-petersen
submitted at: 2026-09-26 10:02:05 UTC
last updated: 2026-09-26 10:02:05 UTC

Witness: 4 independent points log in to add more points →

(-35080, 55774224)
(118568, 8834760)
(3287534/25, 2113798302/125)
(18159341/196, 7004418597/2744)

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), here with (u,v) = (121,558), using the same smooth-support / Nagao-filter search strategy used for curve#1829. PARI/GP 2.17.4 gives exact rank 4 and torsion Z/2Z x Z/4Z. Conductor N = 2428381032.

last edited by jesper-petersen at 2026-09-26 10:02:05 UTC · history