curve #5556

y² = x³ + x² − 65829448296072236277860x + 6353258404939948767284952597170400

a-invariants: [0, 1, 0, -65829448296072236277860, 6353258404939948767284952597170400]
c-invariants: [3159813518211467341337296, -5489215261887074616043467848003249344]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 220660066076915225659080
discriminant (Δ): 820269525861198707613359536251388337280652370830493155680416057760000
Faltings height: 12.0051
naive height: 169.2377
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 29, 47, 53, 61, 79, 271, 751, 883
regulator: 119568.8967570378474780012248365695725581342802170361
submitted by: Daniel Błażewicz
submitted at: 2026-10-23 02:45:46 UTC
last updated: 2026-10-23 02:45:46 UTC

(-293671201125, 18932839667644980)
(172778563870, 11713292448448890)
(19998251709595, 89423666410417846260)
(-1084401308150/81, 61995704294812739230/729)
(775570849117/4, 237140995426403757/8)
(16331164423432/9, 65369606701332450608/27)
(123183662565430/121, 1327307430164493154650/1331)

(u,v) = (191055,679354) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 7.

last edited by Daniel Błażewicz at 2026-10-23 02:45:46 UTC · history