curve #2985

y² + xy + y = x³ − 19331926792477252483376595955959968x + 2155051917219158107902702750444678384523518040310558

a-invariants: [1, 0, 1, -19331926792477252483376595955959968, 2155051917219158107902702750444678384523518040310558]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 17783956777141537113166960878108870
discriminant (Δ): -1543928696699302777937944795258375620155932770150965132064118539683536438493624956450964839805577597750000
Faltings height: 18.8735
naive height: 249.9225
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 43, 71, 109, 137, 211, 239, 269, 1033, 21673, 926921
regulator: 504250417.262729497188993199609760578839668691604203383
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:53 UTC
last updated: 2026-09-26 20:31:53 UTC

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

(72829699846154994, 33666218186691143060657590)
(133043793207834069, 44022957423816284924663515)
(28789059517751423781/529, 432597663752699893798097587625/12167)
(-219382298581854778359/6889, -29921389889588392646510698283995/571787)
(21202357770438572885586/69169, 2878099158326057116855894352397530/18191447)
(6683822577532811601/49, 15552920263257737810862678265/343)
(-547957753729489843382557926/3304525225, -5376972297063063318376455784024662735877/189960632559125)
(2600433823957866438609045726/37819747729, -249513820142902149683328229868315751743045/7354919800101817)

Commentary

2-isogenous image of D-z6old678-067, rational kernel x=1419372898239240055/4. 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 20:31:54 UTC · history