curve #3489

y² + xy = x³ − 2653537278084187322695250426187488x + 52377999300372020618268203510163981734577882350592

a-invariants: [1, 0, 0, -2653537278084187322695250426187488, 52377999300372020618268203510163981734577882350592]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 53400310113364992979902636017282809299958520515445675366550
discriminant (Δ): 10619273669146597002015035527198923831893959421490517772149249777782601827954241999666235572224000000
Faltings height: 18.0392
naive height: 242.4972
primes of bad reduction: 2, 3, 5, 13, 23, 29, 47, 53, 71, 109, 139, 257, 313, 457, 947, 1009, 1117, 2029, 5147, 5581, 6577, 6709, 9173, 16553
regulator: 2964822413952052555619.350218395606473779504030882205180920303201487044
submitted by: Steps Unbounded
submitted at: 2026-09-27 10:22:20 UTC
last updated: 2026-09-27 10:22:20 UTC

Witness: 16 independent points log in to add more points →

(112218440801456419/4, 971239976528185947279215/8)
(-1481731459068902970853232/30283009, 1344303228551963143739090926351273648/166647398527)
(-237495860332869901054478/95199049, 7133618470768715061768367034895253516/928857121093)
(15824662583703134015488/310249, 1218652920596981622735385313112992/172808693)
(256982201004641863503808/1814409, 122633687073103410718965927879420512/2444008923)
(769905107009859655615008/28015849, 67940821897342342501075062812737728/148287888757)
(9494313233032827441547912/126990361, 23596980591930688822421873989795714936/1431054378109)
(10414720650916475257239808/332223529, 155436285395729047545000858015634592/6055438263083)
(56388458021304880319440432/3348821161, 684368328974845746224061965309525690096/193792931765909)
(210482485171580879011823872/6695985241, 89314302788478751769009086476486914656/547925776285789)
(196086490592611908219909070912/6049430474481, 9672399594861375706457598418884264202812896/14878931168384013879)
(309000117219521927362787610688/9852272601889, 2195509381446608871632579240394792956802848/30924638367805055537)
(539635754564083630100319416386/15011897481361, 107905907814189819652670788149323231453904196/58163882017585340359)
(606777413127470774157204679648/29176342689169, 392017413119854012348299619937786086856416288/157596394328001312697)
(139840260922877615799398949014940667/635723733510705316, 50971646319298486405843313372819287545932522406156641/506877061558291002106091864)
(43334586209616100125137682447450784/1592322804383871169, 1134985981039241716381685216515822971204005062590752/2009308735355391110681419553)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=11/5, t=45/901. Found by sweeping t=a/b with the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-27 10:22:20 UTC · history