curve #3105

y² + xy + y = x³ − 27576645475830145153904x + 1729915126735191359803368719620106

a-invariants: [1, 0, 1, -27576645475830145153904, 1729915126735191359803368719620106]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 37170593689294827178345012036890
discriminant (Δ): 49354028070902313727986851150097145882968090702338739044656272284100
Faltings height: 11.7783
naive height: 166.6274
primes of bad reduction: 2, 3, 5, 7, 31, 43, 53, 59, 103, 283, 877, 1291, 1987, 4507, 143677
regulator: 161937424.7065436544686495138371509763343477572993284333770
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:32 UTC
last updated: 2026-09-26 20:37:32 UTC

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

(701564334429, 572439898182939937)
(202305800739, 66564890849359843)
(-182192216332, 26579710796451438)
(278734730043, 125295900226327393)
(162942170538, 39530360962167883)
(1485480872781/25, 2169098217958859489/125)
(57264035646, 18399526044504991)
(155600970168, 34732045952930518)
(-139942098681, 53370960528399307)
(617285342297925/5776, 807886298510133737857/438976)

Commentary

Dujella - Kulesz (2006). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 81). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:37:32 UTC · history