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curve #29

y² + xy + y = x³ − x² − 6494556985832x + 6351402068900282539

a-invariants: [1, -1, 1, -6494556985832, 6351402068900282539]
c-invariants: [311738735319921, -5487609984705535174281]
rank (lower bound): ≥ 8
torsion subgroup: trivial
conductor (N): 72422714076260761748899710
discriminant (Δ): 104878982358622652630973814596384000000
Faltings height: 6.1584
naive height: 100.1196
primes of bad reduction: 2, 3, 5, 37, 151, 251, 5939, 96619896654233
regulator: 192112.135790124168751416663971415608442987571179367034
submitted by: David Renshaw
submitted at: 2026-05-27 22:12:09 UTC
last updated: 2026-07-01 22:56:22 UTC

Witness: 8 independent points log in to add more points →

(1634797, 320450201)
(-2569043, -2464593799)
(1725397, 530370401)
(1536847, 10928201)
(-1037903, -3459834799)
(-947303, -3413266399)
(1536949, 13272713)
(13057, 2503312241)

Commentary

Found by Grunewald - Zimmert (1977). A historical rank ≥ 8 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:32:01 UTC · history

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