curve #3345

y² = x³ + x² − 2269185x + 1300157599

a-invariants: [0, 1, 0, -2269185, 1300157599]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 342404181952
discriminant (Δ): 16700646187670208512
Faltings height: 2.4968
naive height: 55.5184
primes of bad reduction: 2, 7, 31, 71, 113, 439
regulator: 37075.316644477005030213923978327046391216564666739
submitted by: David Renshaw
submitted at: 2026-09-26 21:05:58 UTC
last updated: 2026-09-26 21:05:58 UTC

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

(1049, 8672)
(1469, -33748)
(962, -2899)
(2909, -139012)
(1817, 56384)
(63821/49, 8086756/343)

Commentary

Rationally isogenous to curve#1237 (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 21:05:58 UTC · history