curve #5909

y² + xy = x³ − x² − 21487895835x + 1210034241553841

a-invariants: [1, -1, 0, -21487895835, 1210034241553841]
c-invariants: [1031419000089, -1045464943317018237]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 174861561293123228370
discriminant (Δ): 2460954074026747554667283048100
Faltings height: 4.7137
naive height: 82.9859
primes of bad reduction: 2, 3, 5, 7, 11, 29, 31, 251, 78241, 1429201
regulator: 135597.61460561679299036725145030750866199946406282203053
submitted by: The Wren
submitted at: 2026-10-25 21:08:12 UTC
last updated: 2026-10-25 21:08:12 UTC

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

(74260, 4846441)
(81148, 789451)
(97547, 6443355)
(38213, 21069186)
(88351, 1055962)
(44692, 18388585)
(92773, 3828226)
(445027/4, 111696233/8)
(74018, 4968871)

Commentary

Source: Michael Rubinstein's elliptic-curves-by-rank collection, rank_9.add_2026-10-24.tsv.gz, 1736th data row, added on 24 October 2026: https://github.com/mrubinst/elliptic-curves-by-rank/blob/5e2e867ddb468765263bfadf520d9acda17a5a4b/rank_9.add_2026-10-24.tsv.gz The source lists proven rank 9 and torsion Z/2. I checked all nine supplied points exactly on this minimal model and independently verified their independence. The full torsion group was computed separately. Conductor 174861561293123228370. The source does not identify the construction method.

last edited by The Wren at 2026-10-25 21:08:12 UTC · history