curve #2997

y² + xy = x³ − x² − 1431515389919996945516762018465529x + 20811632200422132045428004921136843090461230381085

a-invariants: [1, -1, 0, -1431515389919996945516762018465529, 20811632200422132045428004921136843090461230381085]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 4538897581549887969893709893633970
discriminant (Δ): 635269273497117346294109213898025615539682023610261922995802423555480790026201770052632342232250000
Faltings height: 17.8468
naive height: 240.6457
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 43, 61, 67, 71, 79, 109, 181, 211, 829, 1479139
regulator: 14203200.8963730818725983609505100445962309762885784123
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:36 UTC
last updated: 2026-09-26 20:32:36 UTC

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

(113265976883053686289129/4981824, 1471063302504282505610439820683361/11119431168)
(798577780426404663920515/27867841, 268109804036391292986124040751929055/147114332639)
(48180710543571511614/2809, 170013422398836183809263813779/148877)
(44619085686831796164654/3523129, 14356794668711921050552205210662611/6612913133)
(142811283164582221, -52240024406178104568223473)
(262322077618318105289806/2569609, -126193259236579049896317198271537947/4119083227)
(37283047612579403238247873044530645660378785885/1843711933809296536780536488464, 912820339366264361276541555950791892948250508416997268775749045880405/2503453909605156783619597903725490073235018688)
(122722593494811496210269924391771947931202443941/5075316903522818427734486732521, -6618442584558105228251082692536504441305933193102282900299835001269812/11433909416683796838880532837342237665073447819)

Commentary

2-isogenous image of D-z6old678-118, rational kernel x=-168849298615036365/4. 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 20:32:36 UTC · history