curve #3148

y² + xy + y = x³ + x² − 21867223320112239016783671995123950489x + 39354506601874758642325518766562884477556802289319976926

a-invariants: [1, 1, 1, -21867223320112239016783671995123950489, 39354506601874758642325518766562884477556802289319976926]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 5656989953868346894737119189064534826653330489
discriminant (Δ): 135900480696836760244965694124816531850167408236219206439848477666969836699436763958718168850197705909320938689
Faltings height: 20.1584
naive height: 269.5478
primes of bad reduction: 3, 7, 11, 17, 41, 43, 59, 73, 89, 179, 181, 197, 293, 353, 433, 443, 1181, 2269, 2389, 2629511
regulator: 23188260756328.1381555001783595698425154962721254094013115413
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:34 UTC
last updated: 2026-09-26 20:39:34 UTC

Witness: 11 independent points log in to add more points →

(24732512710203778780/9, 3301823666656208241591779234/27)
(1761921743490658633, 2509148111321239324296323131)
(163508330617070294453, 2089941357655756156848881887041)
(2298089380222767728, 1112823686752402012572175530)
(7602910627595664572, 17679946510704434260598732925)
(13018573601018243020, 44284362094015236647610890077)
(41516282021171101989/16, 18575917855584889611077597215/64)
(-18467972809008322938016/57121, -92984173365396273958232105339982330/13651919)
(345349833367586271490068/139129, 31532953016429304254615419069916134/51895117)
(8723565315738477727921996/3583449, 5021194902199967569996659253707031070/6783468957)
(277664718631328317393427830/104101209, 71735717309677383678351186541196567239/1062144635427)

Commentary

Voznyy (2022). Source: https://dujella.github.io/tors/z4old9101112.html (curve entry 17). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:39:34 UTC · history