curve #6001

y² = x³ + x² + 62286385x + 319454577889

a-invariants: [0, 1, 0, 62286385, 319454577889]
c-invariants: [-2989746464, -275990816817280]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 533296246689681536
discriminant (Δ): -59545735319715521997883648
Faltings height: 3.6299
naive height: 66.5028
primes of bad reduction: 2, 17, 89, 97, 167, 271, 353, 1777
regulator: 295168.784933759635516537842979952659803150739647491790
submitted by: The Wren
submitted at: 2026-10-27 18:12:40 UTC
last updated: 2026-10-27 18:12:40 UTC

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

(-3351, 270398)
(-2280, 406933)
(-503, 536654)
(10159, 1414492)
(41127, 8511536)
(221581/100, 684363571/1000)
(525856/25, 413378179/125)
(600175/1089, 21380245604/35937)

Commentary

The rationally 2-isogenous partner of the rank-8 curve [0,1,0,-495993705,3489927208999] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 484267. Reduction gives the global minimal model [0,1,0,62286385,319454577889]. PARI/GP returned matching rank bounds [8,8] and the eight rational points listed here. All eight 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 18:12:41 UTC · history