curve #2152

y² = x³ − x² − 7602351178123170260x + 8068044296195949048017420100

a-invariants: [0, -1, 0, -7602351178123170260, 8068044296195949048017420100]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 120256131027065880
discriminant (Δ): 224115658063169355917848543206408746823162562500000000
Faltings height: 9.4403
naive height: 142.0386
primes of bad reduction: 2, 3, 5, 7, 11, 17, 59, 67, 79, 103, 23801
regulator: 135936.13955061186704469321947916750324054501184220
submitted by: jesper-petersen
submitted at: 2026-09-26 12:25:35 UTC
last updated: 2026-09-26 12:25:35 UTC

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

(-1359591260, 126059294962790)
(1597078540, 310650033890)
(14056511590/9, 55694540880440/27)
(30135686144/25, 3200376695822922/125)
(7880439253940/5041, 702884726606095110/357911)
(611313715118920/383161, 39927273276886861250/237176659)

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) = (2278,69125). The search used smooth-support / conductor filtering followed by Nagao scoring and exact rank computation. PARI/GP 2.17.4 gives exact rank 6 and torsion Z/2Z x Z/4Z.

last edited by jesper-petersen at 2026-09-26 12:25:35 UTC · history