curve #3337

y² = x³ − 130791756x + 575619273216

a-invariants: [0, 0, 0, -130791756, 575619273216]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 582587812371744
discriminant (Δ): 54947892753143288082432
Faltings height: 3.3565
naive height: 67.6810
primes of bad reduction: 2, 3, 23, 59, 139, 181, 193, 307
regulator: 1528463.602983096154926302203366593987629158529196731
submitted by: David Renshaw
submitted at: 2026-09-26 21:02:32 UTC
last updated: 2026-09-26 21:02:32 UTC

Witness: 7 independent points log in to add more points →

(24484, -3471404)
(167044/25, -458428/125)
(60196/9, 160588/27)
(190756/25, 18491404/125)
(238276/25, 55176476/125)
(106849/16, 92561/64)
(63652/9, 1783180/27)

Commentary

Rationally isogenous to curve#1242 (isogeny class computed with PARI/GP ellisomat; global minimal model). The witnesses are the images of that curve's witnesses under the isogeny, so they remain independent and the rank bound 7 carries over. Torsion ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:02:32 UTC · history