curve #3083

y² + xy = x³ − 21288624065201247763671805581196335x + 1194654434361719344451732989691125852509802463778297

a-invariants: [1, 0, 0, -21288624065201247763671805581196335, 1194654434361719344451732989691125852509802463778297]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 43591641500121819666966659383620704342130
discriminant (Δ): 929734782426074357090627058392563886539643542927166109839073953738629805708847250042797284627696040000
Faltings height: 18.4918
naive height: 248.7440
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 61, 83, 101, 107, 137, 439, 37463414931029
regulator: 38681297320.91886506575822147608691202236024750277452714836
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:26 UTC
last updated: 2026-09-26 20:36:26 UTC

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

(95872310786639754, -5905522380130812292039947)
(95047692742948704, -5466878926417961718493707)
(90381337358197824, -2977572378369673139892597)
(89289009630428964, -2381876776656314452564227)
(433552460185168371/4, 101370570555549972027098799/8)
(86722230854575638, -820533908388034276043499)
(90771437150444172, -3187867041122986935858297)
(86267723154447894, -384785741033773546770987)
(110583347654293584, 13884348464504283386681883)
(331348242976734266667954/3728761, -15445300149877710636750528203360517/7200237491)

Commentary

2-isogenous image of D-z4old9101112-033, rational kernel x=-41172155077932666. 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 20:36:26 UTC · history