curve #6088

y² + xy + y = x³ + x² − 252979814033828149616633606835402096361712413x − 462321266560434395510360229904992327482519890115183022109083267469

a-invariants: [1, 1, 1, -252979814033828149616633606835402096361712413, -462321266560434395510360229904992327482519890115183022109083267469]
c-invariants: [12143031073623751181598413128099300625362195825, 399445574308215317720860165904861192811035196961057386347557726624375]
rank (lower bound): ≥ 19
torsion subgroup: ℤ/2ℤ
conductor (N): 85846100457142021413369807897710852813676285438536199952814698136832974679317780550
discriminant (Δ): 943849575841156867304711452333072169478500186162127472030386330525094527564154663695214500770426151452281646039509369009284059136000000 ☆ record for ℤ/2ℤ torsion, rank ≥ 19
Faltings height: 24.5909 ☆ record for ℤ/2ℤ torsion, rank ≥ 19
naive height: 318.3393 ☆ record for ℤ/2ℤ torsion, rank ≥ 19
primes of bad reduction: 2, 3, 5, 11, 13, 19, 31, 71, 73, 79, 83, 157, 211, 223, 227, 251, 443, 521, 1627, 3191, 4373, 7109, 8731, 27061, 109829, 118189, 184181, 64459907
regulator: 8258772015982912791327.412174130226459595433105008054949092895994449335736544
submitted by: The Wren
submitted at: 2026-10-29 01:37:14 UTC
last updated: 2026-10-29 01:37:14 UTC

Witness: 19 independent points log in to add more points →

(-243951585533674722217441/25, 129729944187787006341175509767978292/125)
(-48412064267562967596019/25, 17813264318385822059421000562064364/125)
(436501928952635360289119/25, 83236406790167214406970792302999668/125)
(16750667626849762706885, 9043385904339763285878131763132)
(-361750815823269406888369/25, 51317452075945279545844860074449764/125)
(-2994521673691748102665, -518053968650976491504821722926268)
(-127956732324076294260769/25, -104464099820258133087115113075666636/125)
(18022254102948512989015, 912178701961024332858807375151492)
(31330983422122156969015, -4729378605226572690024371666744508)
(-10014385309347949257865, 1032857329342665507322838624712132)
(-6360565789011542744201, 943104110888088791391807566734788)
(16762715767786094852855, 84750932834911395373189293743172)
(92807539700322117156651, 27846594127498834742892090274449688)
(1184613571403225581857271, 1289216472151705599979788214200579780)
(-15610181597903220709745844049/1776889, 2463924075442297697904939211514263943766756/2368593037)
(2097648368209488735969785255/14161, 95510177394116596249425380795101505827548/1685159)
(124931467769404027031132290135/3157729, 40251889445559018793070707339143351283567236/5611284433)
(207158437295224547487034990638815/5776152001, 2655659725938364344603279142258190716246510922892/438993328228001)
(45534635132210116682386961268107495/17697779089, 9716379901021380242405125118017550018891438666780964/2354388645546937)

Commentary

Original specialization/section search by spariev (2026), public Yukon elliptic-rank challenge submission c2120023-a114-4514-ac1b-ff2e5824b97f. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/7520ba9cfcdc31043e79e9baa221b5e59f321170/submission/board/2-19-x552-p18371-q22532.ec . Specialization T=18371/22532 of the rank-nine X552 fibration posted by wgxli with curve#4061: y^2+238(19T-6)(5T-3)xy=x(x+F)(x+G), F=(12T-29)(51T+22)(79T+22)(482T+29), G=256(T-22)(83T-7)(21T-2)(7T+17). The nineteen supplied points have been checked exactly on the minimal model and certified independent modulo torsion by an exact rank-19 F_5 reduction matrix at nineteen good primes.

last edited by The Wren at 2026-10-29 01:37:14 UTC · history