curve #6006

y² = x³ + x² − 348523671089x + 79159953450725631

a-invariants: [0, 1, 0, -348523671089, 79159953450725631]
c-invariants: [16729136212288, -68394300156244218880]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/4ℤ
conductor (N): 201673312466208
discriminant (Δ): 2374222478046588848823402578817024
Faltings height: 5.3562
naive height: 91.3445
primes of bad reduction: 2, 3, 11, 17, 83, 229, 239, 2473
regulator: 14324.328827194646035939397725305159997484422893453
submitted by: The Wren
submitted at: 2026-10-27 18:45:47 UTC
last updated: 2026-10-27 18:45:47 UTC

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

(-597285, 272483436)
(241771, 95023092)
(291235, 48576396)
(346882, 1672845)
(1439053/4, 147097005/8)
(3014995/9, 3565988/27)

Commentary

Rationally 2-isogenous to the rank-6 curve [0,1,0,-25527082394,782570227763016] in Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2xZ2.add_2026-10-24.tsv.gz, source index 114 (115th data row). The source points were transported through the degree-2 isogeny to the minimal model [0,1,0,-348523671089,79159953450725631]. Using these points, PARI/GP returned matching rank bounds [6,6] and the six independent points listed here. All six were checked exactly on this model. Rational torsion Z/4Z. Source: https://github.com/mrubinst/elliptic-curves-by-rank/blob/5e2e867ddb468765263bfadf520d9acda17a5a4b/torsion/torsion_Z2xZ2.add_2026-10-24.tsv.gz

last edited by The Wren at 2026-10-27 18:45:48 UTC · history