curve #3349

y² + xy = x³ − x² − 1500752161260x + 707639060345594706

a-invariants: [1, -1, 0, -1500752161260, 707639060345594706]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/6ℤ
conductor (N): 4149942570
discriminant (Δ): 12854631399414927145471138934070
Faltings height: 5.4671
naive height: 95.7246
primes of bad reduction: 2, 3, 5, 19, 37, 107, 613
regulator: 3736.096197131434867365898516495908508096572721
submitted by: David Renshaw
submitted at: 2026-09-26 21:09:25 UTC
last updated: 2026-09-26 21:09:25 UTC

Witness: 4 independent points log in to add more points →

(749415, 61601421)
(19361095/4, -82711160373/8)
(708517, 1436030)
(9683402665/13689, -521482247623/1601613)

Commentary

Rationally isogenous to curve#1612 (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 4 carries over. Torsion ℤ/6ℤ.

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