curve #6126

y² = x³ + x² − 443764949643304409918775231259576671865508x + 113756777690381075649505886186659202715456527446363275538268988

a-invariants: [0, 1, 0, -443764949643304409918775231259576671865508, 113756777690381075649505886186659202715456527446363275538268988]
c-invariants: [21300717582878611676101211100459680249544400, -98285855924489249361300889970770822816211046980260628146561672000]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 120096546094927573418647101461067778384553568121916552400
discriminant (Δ): 2583471425475989454749288341075684399458487180372264829147132295120007381789076890045089257083466209292333656552460100000000
Faltings height: 22.6644
naive height: 299.3019
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 29, 31, 47, 67, 79, 103, 181, 193, 277, 389, 3347, 4973, 10193, 82847, 85453, 1891367
regulator: 8308438863222314.059214976492304781325919698427600300163438438220
submitted by: The Wren
submitted at: 2026-10-29 03:19:13 UTC
last updated: 2026-10-29 03:19:13 UTC

Witness: 13 independent points log in to add more points →

(121216524023914557919, 7857876394053858288137584902984)
(-672652634523615850683384443/1094116, 14211789669489244586927743645992245820417/1144445336)
(-2104885406214386819963/4, 113577459157810587367879642731375/8)
(6390918688269200868673/16, 30734521077367797509941602206775/64)
(49822903902592762541758/121, 1223127045932820308224448521779000/1331)
(244826293100491339689637/4, 121132720013938167073553750292271425/8)
(592791520545436121346418/3481, 1348716204515602630334518271298282000/205379)
(21824252962157705015211598/361201, 2026717383110737105489010076446855175000/217081801)
(7282176570556979578405309479/22231225, 197280008455787203107503327724204167651592/104820225875)
(27912292185265431967429647022/1138489, 4661595193463855188313652608662246659126000/1214767763)
(78071604486137175709538033323/98982601, 15707940563468265122441079976925581535301750/984777897349)
(5503499379785191384494005664142/15059462089, 1154142373452788072223748709791592158113073200/1848052009175813)
(167596626008021319613919888089593358/1105208814791761, 259678082565262663413222872910641775841374347780152200/36742314486140175873241)

Commentary

Source parameter/search by bufferhe4d (2026), public Yukon submission 15009775-297e-4a19-9179-4a455b778310: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/6837b3bcd53a73fb5af3e271d91dec3148236631/submission/search/x400a-opus/xlog.txt . The source log records rank13 at T=74052/85453 in the rank-five X400a family posted by wgxli with curve#4252: y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q). The published section code credits 0xArion submission7e9f6673-cf5f-4cb9-82b4-a22add86d720. An initial local PARI/GP effort0 replay seeded by its four selected sections returned bounds[13,13] but only twelve points, reproducing the source gap. One effort1 replay seeded by those twelve points recovered the thirteen displayed points with bounds[13,13]. Every coordinate was checked exactly on the minimal model; independence modulo torsion was separately certified by an exact rank-13 F_5 reduction matrix at thirteen good primes.

last edited by The Wren at 2026-10-29 03:19:13 UTC · history