curve #325

y² = x³ + x² − 10144481942155625929509103657049467471830992522065306476640x + 380216286633517255601187823421134146777992443645481380690941792226978359266852679415988

a-invariants: [0, 1, 0, -10144481942155625929509103657049467471830992522065306476640, 380216286633517255601187823421134146777992443645481380690941792226978359266852679415988]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/4ℤ
conductor (N): 349432175747971837118686050547170875372336606890306428283195130479920
naive height: 412.3064
Faltings height: 32.2786
discriminant (Δ): 4362494941807848704026559610365890975197207337251115281214370363978888613665527680012511255596630140495384679977316682533728496642675860125703755399138653218414361000000000000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 101, 131, 139, 149, 157, 179, 653, 829, 953, 1307, 4019, 7207, 29287, 44249, 121178117, 3622521358583
regulator: 34430367495376806.82338143341906110207956966312996387696791856339
submitted by: Jack Cheng
submitted at: 2026-08-24 13:51:15 UTC
last updated: 2026-08-24 13:51:15 UTC

Witness: 13 independent points

(98797428465597921854659966106, 18501960057154462753965244476587904114894120)
(70608658515713292144980020271, 3994161829276020893108640059675171105221650)
(46520024909903078786877295346, 2994880885411491385752108474380838770935200)
(-57720943255918350284108413429, 27811078111529201124489016436115090724245000)
(-764989076382421108633641129991/64, 11445977979062421025994112960416678952063818835/512)
(25388143782459322232605563700586/841, 245694376936377578672166940167101852223554925000/24389)
(-86863614263322156691238785918, 24616898901745765937901875786280291487546560)
(7628383854510617586079154579444/169, 8301959381108234271133955319370381943077235670/2197)
(-68409654340863990558041719868, 27459930105477512767395593422236922524531870)
(195044948131617038220488149956412394/5139289, 82283431133081348768684829999230925995889629338125000/11650768163)
(3414368457420016094738135714485154/76729, 87404877035492432837436131608169349969327057276480/21253933)
(16335531540912726198847501141979/49, 63292804053841425659456731624558368649977060000/343)
(-84774382922118921353920367428877473/877969, 17634527581279166417560235769165452719051524269905614/822656953)

Commentary

Li's 2026 elliptic curve over Q with torsion Z/4Z and rank at least 13, with all 13 published independent witnesses. Source and maintained record table: https://web.math.pmf.unizg.hr/~duje/tors/z4.html. This is a documented existing world-record curve newly added to this leaderboard, not a claim of a newly discovered world rank record. Exact local preflight re-certifies every witness, independence modulo torsion, the global minimal model, torsion, and the complete bad-prime set.

last edited by Jack Cheng at 2026-08-24 13:51:15 UTC · history