curve #319

y² + xy = x³ − x² − 1478818379630960182018543975144238479079598870400357903794x + 21612333371564362906227820064846376685227380006732368388851217276076518563786908263808

a-invariants: [1, -1, 0, -1478818379630960182018543975144238479079598870400357903794, 21612333371564362906227820064846376685227380006732368388851217276076518563786908263808]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 50641610943684620657478317246146550331473995146991135940486156432435703726760290
naive height: 406.5294
Faltings height: 31.7529
discriminant (Δ): 5193990139081890317459401622566217412295259126704665225250680699618492464914335380075351240164897155445181855895609689483201815665345724128042696272338696704023011289062500
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 29, 31, 43, 53, 59, 67, 79, 101, 107, 109, 167, 233, 281, 307, 317, 419, 571, 761, 863, 1021, 1531, 1543, 1993, 3299, 3727, 3821, 16417, 19273
regulator: 75452645054176208.282145425803742497370186324948193253851984275183296
submitted by: Jack Cheng
submitted at: 2026-08-24 11:31:13 UTC
last updated: 2026-08-24 11:31:13 UTC

Witness: 15 independent points

(11183035360176737872066462162, 2544250119932764965168254926681650601072814)
(-14387255078284602255172684728, 6317468477138182570583042837408855896541364)
(19438262913631476179745098142, 459721721339879324467548812793427818227814)
(55823249110172914794349281022, 10631004522185171552847940461954446457166364)
(32572289692579628851611993897, 2828698563670717423033345745417611680180114)
(24588470142629085616740326022, 341270246931310610931397778424345615952614)
(19982025847826040867579006207, 202476372279358043207285606116969759878864)
(24988563558081184745544566922, 512203846743830750521268371377174156493614)
(19409175712041011823546618522, 470570514278064910180708844073966212677614)
(15343558255011251183303194179, 1591933177167506122828050814070561825165145)
(24354530036362110771030330897, 205253458293167433649155238598749814431989)
(40442125414745602621502494162, 5286919745747422460169589189761390362993024)
(24943293985561753095203275747, 494620041717645188762260334392615608575364)
(266053292109858198648954299847, 135869710793553829756471002267336612487527489)
(1872527673822446130016910594397, 2561836921226411679741258783032933091672508614)

Commentary

The current rank record for elliptic curves over Q with torsion Z/2Z x Z/2Z, found by Noam Elkies in 2009: rank at least 15 with the 15 published independent witnesses shown here. Source and maintained record table: https://web.math.pmf.unizg.hr/~duje/tors/z2z2.html. The leaderboard's exact Cremona/Brumer verifier re-certifies every witness and independence modulo the full rational 2-torsion subgroup. Native PARI ellglobalred/elltors independently confirms the global minimal model, torsion [2,2], complete bad-prime set, and conductor. This is a documented existing world-record curve newly added to this leaderboard, not a claim of a new world rank record.

last edited by Jack Cheng at 2026-08-24 11:31:13 UTC · history