curve #3168

y² + xy + y = x³ − 81089552755699140908030202688368478038x + 281263936268193464077589295595406966386314342335769150988

a-invariants: [1, 0, 1, -81089552755699140908030202688368478038, 281263936268193464077589295595406966386314342335769150988]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 11832888960533551059315269623085924821036068375150270
discriminant (Δ): -50102147809672832424664968556447527556796330304757314437189643661176333559733455907337999555790043788121188312500
Faltings height: 20.5523
naive height: 273.4809
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 89, 2609, 22093, 22335427, 13141591273
regulator: 38417162528.725489625548664392733412186899778557813610069131037
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:33 UTC
last updated: 2026-09-26 20:40:33 UTC

Witness: 12 independent points log in to add more points →

(5741172515907340367, 2224852147461708059604916497)
(-6283260873451047176, -23296171078888629764646328180)
(-4377070887290532121, -23501903154705392080910055465)
(2577560963520055004, 9453862857802006685518121535)
(2012165467991458804, 11235889541275053343944971060)
(5052438574023597248, 733645881558773670043750083)
(2616398712495877619, 9328022441806695951924810555)
(3958074780969203204, 4723772959038114089554236135)
(6205498548772362149, 4126195518911771323234166295)
(-5276925921661264561, -23711315833456709412101226735)
(5589234149317131623, 1624971859309398399443882583)
(3832164828660767909, 5176158037521448409935173015)

Commentary

Eroshkin (2006). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 11). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:33 UTC · history