curve #3695

y² = x³ − 101877280012210949799726396x + 389230831065122113578184713346329713920

a-invariants: [0, 0, 0, -101877280012210949799726396, 389230831065122113578184713346329713920]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 282976117803100217896404857655264
discriminant (Δ): 2224188884104094034322632122718427516963876101332038285802870000202356187131904
Faltings height: 13.8268
naive height: 191.2710
primes of bad reduction: 2, 3, 7, 11, 29, 31, 43, 59, 61, 67, 83, 107, 113, 151, 277, 4397, 7417
regulator: 13112560845.10097020991104734920504641138557178642649554564
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:44:55 UTC
last updated: 2026-09-30 01:44:55 UTC

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

(3831814776653, 7424133032762786403)
(-30460542435926462070/3301489, 139888659168364687981069672660/5998805513)
(-282430881219454/25, 1239482402141023478844/125)
(-31443375429783/4, 212313451224433117909/8)
(304275607581588/121, 16243894084540941931532/1331)
(12024313702738769/1600, 443624956529479789291353/64000)
(38167047320189066/13225, 16608361898303355519374364/1520875)
(38791388080448720/841, 7470512902108147173803640/24389)
(101996256370422153/20164, 5223284915079830056569797/2863288)
(2106204613179794186/727609, 6758616905594134705701383940/620650477)

Commentary

Specialization at t = 565/2324 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 2026-09-30 01:44:55 UTC · history