curve #3681

y² + xy + y = x³ − x² − 326652986756280439298808750317x + 71297354488784251332784398142383987320276309

a-invariants: [1, -1, 1, -326652986756280439298808750317, 71297354488784251332784398142383987320276309]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 12441034926946153882022260307768910
discriminant (Δ): 34702249000017957985375764018401853182143737643590123594744164048166126339234201600000000
Faltings height: 15.8120
naive height: 215.4897
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 31, 37, 47, 53, 167, 2591, 16141, 18911, 6551417
regulator: 71890020.25086436490078632076921300186039244306300194839939
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:32:09 UTC
last updated: 2026-09-29 19:32:09 UTC

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

(14961583349967, 8149445233517627397016)
(776823972263967, 16921115704804200093016)
(857105769190287, 20517725152225349774296)
(939992583429967, 24388732787266259253016)
(1439512375013967, 50833334280913319493016)
(1831090722272967, 74917405560588049278016)
(2562103395993967, 126700365977701273973016)
(2751534231093967, 141436213119256039173016)
(56533477280421967, 13441150332264041869125016)
(858321224143658247/841, 690250012202797210556607224/24389)

Commentary

Specialization at t = 463/1955 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:32:09 UTC · history