curve #2983

y² + xy + y = x³ − x² − 829178471835616301952748639246733288x + 290440349789179119333415743475298029821957627286914731

a-invariants: [1, -1, 1, -829178471835616301952748639246733288, 290440349789179119333415743475298029821957627286914731]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 279518233526010740846353904622630
discriminant (Δ): 44195162062651873570853949323245089183220239193598113991095773955965982190351727346615993053191916328550400
Faltings height: 19.3997
naive height: 259.7308
primes of bad reduction: 2, 3, 5, 11, 13, 17, 29, 31, 37, 59, 67, 71, 101, 109, 137, 157, 331, 677, 2579
regulator: 16088966.2005044102984960496513716516788480001488705707
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:50 UTC
last updated: 2026-09-26 20:31:50 UTC

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

(5467360413203851896122109/10195249, -12310033074503954543458870065882911/32553430057)
(14786236441032287237349/26569, -159032130137384990559939583230221/4330747)
(2722531763216899927671/3025, 86961227433921627806549639001241/166375)
(-23688658781097188585715/101761, 22274722978835723289599657255912263/32461759)
(2763283742551687245, -4370228026214791828939132103)
(-92195778694216032330555/109561, 22714172299787630101931808528924187/36264691)
(108967738861716748534498245/510895609, 4054725718042739814561331794280485401659/11547773450227)
(1787345929950214088169177045/1710897769, -53162625987125472676454562237627994032781/70767864419147)

Commentary

2-isogenous image of D-z6old678-054, rational kernel x=-257564734993073139. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:31:50 UTC · history