curve #2972

y² + xy = x³ − 518090659353812744758857689461949066x + 352144435623693808174093901816494401103766251825982500

a-invariants: [1, 0, 0, -518090659353812744758857689461949066, 352144435623693808174093901816494401103766251825982500]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 2059735484183469504222249000949231390
discriminant (Δ): -44670315256660085289614946510325585562907075236420454898967729067161240142828390265413196085845580078016102400
Faltings height: 19.7269
naive height: 260.1149
primes of bad reduction: 2, 5, 7, 13, 19, 31, 37, 41, 47, 59, 61, 67, 101, 139, 461, 1231, 1867, 15027101
regulator: 71205929.5240556216203660310872410911243294583281651870
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:05 UTC
last updated: 2026-09-26 20:31:05 UTC

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

(220882760547855678904/441, 4326993703625014680685542454970/9261)
(656892175423109748, -543386165165085342991160474)
(36462052065818001080676/100489, 14664678745856608094794122214195438/31855013)
(-195452771575426412, -667787425648837458940452954)
(4753201458650490400284052/16297369, -31266990169747230761963535682283883762/65792478653)
(8974988097547702847441428/241081, -26882590415070742185921196331936848814/118370771)
(111766661434996149069347056/36517849, -1155895512438792344808199162085040688098/220677361507)
(167728567372966492852/1089, -18879805775339694980913402292378/35937)

Commentary

2-isogenous image of D-z6old678-056, rational kernel x=7551961523951506015/4. 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 20:31:05 UTC · history