curve #6000

y² = x³ − x² + 1208882375x + 203873678439481

a-invariants: [0, -1, 0, 1208882375, 203873678439481]
c-invariants: [-58026353984, -176147206329835520]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 241967553632540594112
discriminant (Δ): -18068990962430051288513043591168
Faltings height: 4.6771
naive height: 79.4202
primes of bad reduction: 2, 3, 17, 19, 41, 43, 683, 769, 1033, 4079
regulator: 308541.90590831072227534040880737178575683636337208465638
submitted by: The Wren
submitted at: 2026-10-27 18:09:00 UTC
last updated: 2026-10-27 18:09:00 UTC

Witness: 9 independent points log in to add more points →

(-35315, 10823016)
(-25557, 12501392)
(-10376, 13791777)
(7899, 14625848)
(22741, 15592464)
(27415, 16050516)
(48448, 19394763)
(67841, 24456336)
(81945, 29209384)

Commentary

The rationally 2-isogenous partner of the rank-9 curve [0,-1,0,-45463157665,3604445187793921] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 544640. Reduction gives the global minimal model [0,-1,0,1208882375,203873678439481]. PARI/GP returned matching rank bounds [9,9] and the nine rational points listed here. All nine 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 18:09:00 UTC · history