curve #2123

y² = x³ − x² − 1295200614248800x + 10024622212124665600000

a-invariants: [0, -1, 0, -1295200614248800, 10024622212124665600000]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 63897470352240
discriminant (Δ): 95643443473701972219342591688748275569469440000
Faltings height: 7.7130
naive height: 116.0059
primes of bad reduction: 2, 3, 5, 7, 23, 31, 41, 271, 4801
regulator: 14352.2997251386372781408093898120145446010129201
submitted by: jesper-petersen
submitted at: 2026-09-26 10:03:45 UTC
last updated: 2026-09-26 10:03:45 UTC

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

(52433250, 293708919250)
(-469672062525680/37075921, 35263495560386170733040/225755282969)
(-150353315/4, 598727020815/8)
(105691600/121, 125523612068400/1331)
(195168929250/49, 86218075534945250/343)

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) = (4801,8401). Here v-u = 3600 = 60^2. The search used the same smooth-support / Nagao-filter strategy used for curve#1829. PARI/GP 2.17.4 gives exact rank 5 and torsion Z/2Z x Z/4Z. Conductor N = 63897470352240. This substantially improves the conductor of curve#1829 that I previously submitted.

last edited by jesper-petersen at 2026-09-26 10:03:45 UTC · history