curve #3016

y² + xy = x³ + 295194034822759569331514389334x + 534534115609801864257376534758175549756242020

a-invariants: [1, 0, 0, 295194034822759569331514389334, 534534115609801864257376534758175549756242020]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 2958865796524431023234757779070
discriminant (Δ): -125080219586307745595131541638458069559166852117722698172166098995431101366098847179464103360
Faltings height: 16.3516
naive height: 219.5031
primes of bad reduction: 2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 43, 53, 59, 71, 73, 12569, 3131981
regulator: 752657465.959773324801349246137556600239257430899817981
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:25 UTC
last updated: 2026-09-26 20:33:25 UTC

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

(10636318943609816, 1098623995752019639133810)
(27558445579082966, 4575852370271083746147674)
(36511571628856480/9, 7066546611864758671832050/27)
(13454002593511539896/134689, -1174976455453705143143914304470/49430863)
(12479792326037108720/22801, 100971410823712337126889820730/3442951)
(-4127709764196076826/10201, -19243072248512152617417521614/1030301)
(5217928767668516, -379660689846054020286226)
(8399929116702949664, -24345204830416552880818193674)

Commentary

2-isogenous image of D-z6old678-030, rational kernel x=345678105543668. 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 20:33:25 UTC · history