curve #2869

y² + xy = x³ − 66947013578300710021560x + 13202117236838613534512612222145600

a-invariants: [1, 0, 0, -66947013578300710021560, 13202117236838613534512612222145600]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 3673224997035980575470
discriminant (Δ): -56092628853366673399152173840721136214004124698027738359261204544000000
Faltings height: 12.2661
naive height: 170.6545
primes of bad reduction: 2, 3, 5, 23, 41, 43, 79, 109, 139, 1061, 2377729
regulator: 911949.95097188486849274806495046702179756793146659
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:26:07 UTC
last updated: 2026-09-26 20:26:07 UTC

Witness: 6 independent points log in to add more points →

(12008964230250/121, 115539495422600008710/1331)
(191471517688320/1369, 4107159899026633817640/50653)
(174597191250, 82678862670862410)
(-880426296397440/4489, 41225323911384506238360/300763)
(-341738728226880/14641, 215170127300679481504440/1771561)
(-12774845320300800/63001, -2147331886985192829797160/15813251)

Commentary

2-isogenous image of D-z6old678-162, rational kernel x=163751652720. 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 20:26:07 UTC · history