curve #3474

y² + xy + y = x³ + x² − 314208155661995x − 337913352206463230080

a-invariants: [1, 1, 1, -314208155661995, -337913352206463230080]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 10196321216235
discriminant (Δ): 1936000236295596876521355341643846645875390625
Faltings height: 7.3798
naive height: 111.7568
primes of bad reduction: 3, 5, 7, 11, 17, 19, 59, 149, 3109
regulator: 2183.80024789901072662966740852258475523428790334
submitted by: jesper-petersen
submitted at: 2026-09-27 08:01:24 UTC
last updated: 2026-09-27 08:01:24 UTC

Witness: 5 independent points log in to add more points →

(-4026630, 29361847870)
(-2681607, 22032817543)
(19153598, 25885701263)
(92389767/4, 549931834979/8)
(-25719307/16, 817171591957/64)

Commentary

Found by a conductor-oriented search in the standard Z/2Z x Z/4Z family \(y^2=x(x+u^2)(x+v^2)\), using conditions \(u+v=d w^2\) with small squarefree \(d\). Here \((u,v)=(8791,11900)\) and \(u+v=20691=19\cdot33^2\). PARI/GP 2.17.4 gives exact rank 5 and torsion Z/2Z x Z/4Z. Conductor \(N=10196321216235\). This improves my preceding rank-5 submission of conductor \(38019384450552\).

last edited by jesper-petersen at 2026-09-27 08:01:24 UTC · history