curve #5993

y² + xy + y = x³ + 4673525847x + 166564735530872

a-invariants: [1, 0, 1, 4673525847, 166564735530872]
c-invariants: [-224329240679, -143911595004812605]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 53548916157398492398
discriminant (Δ): -18518289350613785040271651676188
Faltings height: 4.6848
naive height: 79.0160
primes of bad reduction: 2, 17, 47, 59, 103, 149, 191, 239, 463
regulator: 335822.91740667883093747186088352237597124094879381879797
submitted by: The Wren
submitted at: 2026-10-27 17:24:40 UTC
last updated: 2026-10-27 17:24:40 UTC

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

(-21122, 7654335)
(-1711, 12593048)
(34338, 19153966)
(43221, 21175060)
(485735/4, 401491229/8)
(268981, 144380758)
(-2408699/121, 10797005810/1331)
(1353224/25, 3002167472/125)
(28060105/256, 182757975571/4096)

Commentary

The rationally 2-isogenous partner of [1,0,1,-32115842143,1707229888216092] from Michael Rubinstein’s elliptic-curves-by-rank repository, torsion_Z2.add_2026-10-24.part3.tsv.gz, data row 544594. Reduction gives the global minimal model [1,0,1,4673525847,166564735530872]. 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 17:24:40 UTC · history