curve #3086

y² + xy = x³ − 2349435075187615767950697058757804x + 43721279197334026347875973048560460636469040598864

a-invariants: [1, 0, 0, -2349435075187615767950697058757804, 43721279197334026347875973048560460636469040598864]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 86459235830591416435905931905748037526
discriminant (Δ): 4195432898049177133335195048046525048583929454311274641232013549607663097627936773262100490817799744
Faltings height: 17.9862
naive height: 242.1321
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 73, 1367, 1931, 5905499, 29036999
regulator: 27634049171.65227236786783995102147588761015388047312706237
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:37 UTC
last updated: 2026-09-26 20:36:37 UTC

Witness: 10 independent points log in to add more points →

(30015344123370478, -493586123569157845227344)
(53236660073998228, 8338192565460008471136916)
(31228429798505578, -898074995982478155112874)
(60707015819686636, 11172295527706439370767092)
(34992341074424386, 2087080556277450811602622)
(1070068163212348942, 1105805950210327823586399664)
(109451982769055977978/2809, -501218882642242095414014483392/148877)
(32211407654827228, -1210129159170633110988944)
(81272949236295721470681028/2052180601, -332479389825430560125276657596391239484/92965833405901)
(351646291300680028, 206640783436843805124714736)

Commentary

2-isogenous image of D-z4old9101112-029, rational kernel x=-13413637921750568. Source: https://dujella.github.io/tors/z4old9101112.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 20:36:37 UTC · history