curve #3039

y² + xy = x³ − 26063688229376963590087835932610x + 50137806027135463507471054481447125468330844100

a-invariants: [1, 0, 0, -26063688229376963590087835932610, 50137806027135463507471054481447125468330844100]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1787522945026386872439824060190
discriminant (Δ): 47189063435064566192444373364955256022383533696190718925593009747696137495697251989927362560000
Faltings height: 16.9508
naive height: 228.6279
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 31, 43, 53, 67, 71, 79, 89, 269, 2129, 65129
regulator: 2315386.00748038473679747882929878122023278042626632556
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:34:29 UTC
last updated: 2026-09-26 20:34:29 UTC

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

(1481665335755740, 121543778319648659823190)
(1590633017588260, 112714350004850071796590)
(-937585297049084, -271570501176183017692178)
(1432584187704766, 125457032879934936033442)
(2407660419575260, 36634552150980951701590)
(2523242437805380, 20920363161016029015790)
(-211328498033180, -235873627583674686622610)
(-1551632189314490, -294692062672567991257160)

Commentary

Dujella - Peral (2019). Source: https://dujella.github.io/tors/z2z4old678.html (curve entry 17). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:34:29 UTC · history