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curve #4048

y² = x³ − x² − 1189552289x + 15110558153889

a-invariants: [0, -1, 0, -1189552289, 15110558153889]
c-invariants: [57098509888, -13055179653900800]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 467503056192 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 6
discriminant (Δ): 9095555819753370641852596224
Faltings height: 4.1243
naive height: 74.3041
primes of bad reduction: 2, 3, 17, 47, 89, 97, 353
regulator: 1058.0911966346140893840100881902484094145281369521
submitted by: Michael Rubinstein
submitted at: 2026-10-07 22:24:14 UTC
last updated: 2026-10-07 22:24:14 UTC

Witness: 6 independent points log in to add more points →

(253889, 126801504)
(11809, 1646144)
(173369/25, 335287872/125)
(-36335, 3218848)
(375557/4, 216306045/8)
(-5392, 4622535)

Commentary

Torsion Z/2+Z/2. From Michael Rubinstein's github repository: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by Michael Rubinstein at 2026-10-07 22:24:14 UTC · history

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