curve #6002

y² = x³ + x² − 115475439x − 393426745775

a-invariants: [0, 1, 0, -115475439, -393426745775]
c-invariants: [5542821088, 339887451423104]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 362387068339061632
discriminant (Δ): 31694367923515373587857152
Faltings height: 3.6113
naive height: 67.3073
primes of bad reduction: 2, 41, 73, 97, 223, 353, 1697
regulator: 235329.953935739568658153809744197287189913003087764140
submitted by: The Wren
submitted at: 2026-10-27 18:15:42 UTC
last updated: 2026-10-27 18:15:42 UTC

Witness: 8 independent points log in to add more points →

(-6528, 286793)
(-4831, 227398)
(12799, 474792)
(46079, 9598232)
(76625, 20991890)
(50253/4, 2981629/8)
(90981/4, 23661271/8)
(173737/9, 57728546/27)

Commentary

Rationally 2-isogenous to the rank-8 curve [0,1,0,-552775369,4641732108231] in Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, source index 484189. The eight source points were transported through the degree-2 isogeny to the minimal model [0,1,0,-115475439,-393426745775]. PARI/GP returned matching rank bounds [8,8]. All eight points were checked exactly on this model. Rational torsion Z/2Z. Source: https://github.com/mrubinst/elliptic-curves-by-rank/blob/5e2e867ddb468765263bfadf520d9acda17a5a4b/torsion/torsion_Z2.add_2026-10-24.part3.tsv.gz

last edited by The Wren at 2026-10-27 18:15:43 UTC · history