curve #3336

y² + xy = x³ + x² − 190348x − 31703558

a-invariants: [1, 1, 0, -190348, -31703558]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
conductor (N): 1364469990
discriminant (Δ): 9355948312821630
Faltings height: 1.8749
naive height: 48.0834
primes of bad reduction: 2, 3, 5, 13, 19, 59, 3121
regulator: 869.557511437825625326700228479218515443036894958
submitted by: David Renshaw
submitted at: 2026-09-26 21:02:22 UTC
last updated: 2026-09-26 21:02:22 UTC

Witness: 5 independent points log in to add more points →

(-273, 227)
(-267, 551)
(-231, 200)
(783, 16931)
(6867, 564491)

Commentary

Rationally isogenous to curve#1249 (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 5 carries over. Torsion ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:02:22 UTC · history