curve #3045

y² + xy + y = x³ − 255549243730155982684226561133x + 49636370655370589957824311621959424162853556

a-invariants: [1, 0, 1, -255549243730155982684226561133, 49636370655370589957824311621959424162853556]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1373748530431186186949942270
discriminant (Δ): 3731644350991827475027681266057847730313914350140945824833224192443012863374180664062500
Faltings height: 15.6904
naive height: 214.7532
primes of bad reduction: 2, 5, 7, 11, 13, 17, 19, 23, 41, 47, 61, 73, 101, 389, 1031, 53147
regulator: 10033574.9216863613579236543673892437529705663596891295
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:34:43 UTC
last updated: 2026-09-26 20:34:43 UTC

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

(1080750704793079/4, -4484196680226685555303/8)
(959688212274610/9, 131163000626811758837134/27)
(-264314926319210, -9935601524910475254458)
(20192307468522885, -2868428937468835159972478)
(1716226683971050810/289, -2240446377396472166097004029/4913)
(94111774828761157, -28870858212331133841922632)
(1881844567162510/9, 62417489003083234739434/27)
(101738664853518864015/863041, -3687196420813833687736167530512/801765089)

Commentary

2-isogenous image of D-z6old678-065, rational kernel x=291804693239683. 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 20:34:44 UTC · history