curve #3346

y² = x³ + x² − 6450505x + 6303413319

a-invariants: [0, 1, 0, -6450505, 6303413319]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 1188797217664
discriminant (Δ): 1172211118883151872
Faltings height: 2.5508
naive height: 58.6526
primes of bad reduction: 2, 7, 31, 71, 617, 977
regulator: 138443.71593264582625010765584848628980920521131680
submitted by: David Renshaw
submitted at: 2026-09-26 21:06:27 UTC
last updated: 2026-09-26 21:06:27 UTC

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

(1834, -25403)
(1474, 257)
(45298/9, 8596117/27)
(72349/36, -8255627/216)
(54709/36, 767233/216)
(1403833/784, 489615307/21952)

Commentary

Rationally isogenous to curve#1233 (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 21:06:27 UTC · history