curve #3340

y² = x³ − x² − 777200x − 194861424

a-invariants: [0, -1, 0, -777200, -194861424]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 778285688496
discriminant (Δ): 13598356980254303232
Faltings height: 2.3802
naive height: 52.3040
primes of bad reduction: 2, 3, 17, 47, 53, 101, 223
regulator: 28885.501139708944481952895560774847647876370170461
submitted by: David Renshaw
submitted at: 2026-09-26 21:03:11 UTC
last updated: 2026-09-26 21:03:11 UTC

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

(-699, 2526)
(-347, 5738)
(1056, 12684)
(1408, 38732)
(2208, 94068)
(8896, 834764)

Commentary

Rationally isogenous to curve#1241 (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 6 carries over. Torsion ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:03:11 UTC · history