curve #5904

y² = x³ − x² − 50441588805x + 4358876817634686

a-invariants: [0, -1, 0, -50441588805, 4358876817634686]
c-invariants: [2421196262656, -3766055043258792800]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 318907156680908982348
discriminant (Δ): 5986794036247879118577848469072
Faltings height: 4.8667
naive height: 85.5458
primes of bad reduction: 2, 3, 17, 43, 53, 149, 229, 271, 281, 293
regulator: 309833.41522200896211942488436768290358058258139904568420
submitted by: The Wren
submitted at: 2026-10-25 20:55:32 UTC
last updated: 2026-10-25 20:55:32 UTC

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

(18723, 58489503)
(134049, 2441067)
(125363, 2354207)
(1835221/9, -1363962241/27)
(745251, 616991391)
(1014529/9, 276955687/27)
(113279, -9924833)
(115989, 8285283)
(135539, 3466783)

Commentary

Source: Michael Rubinstein's elliptic-curves-by-rank collection, rank_9.add_2026-10-24.tsv.gz, 1880th 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 318907156680908982348. The source does not identify the construction method.

last edited by The Wren at 2026-10-25 20:55:32 UTC · history