curve #6361

y² + xy = x³ − x² − 3397195943464662755034x + 85379185939739385002961204029024

a-invariants: [1, -1, 0, -3397195943464662755034, 85379185939739385002961204029024]
c-invariants: [163065405286303812241641, -73767616651201034318770113125989365]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 8062007740848746854165180262204606580281604562587083681994 ★ record for rank ≥ 20
discriminant (Δ): -639872057346046317991296150828830747250399382402362104255286391068
Faltings height: 11.3570
naive height: 160.5725
primes of bad reduction: 2, 3, 13, 223, 727, 25579, 8308153608824869680085984773076870626144186299
regulator: 162473468679904806290617743974.185500605370291025941817479054998284306660539342
submitted by: The Wren
submitted at: 2026-11-01 14:05:10 UTC
last updated: 2026-11-01 14:05:10 UTC

Witness: 20 independent points log in to add more points →

(1177018425061938580087/216056702761, 822239962936992403836187901026232/100427260520665259)
(2008716997211809688767/29265287041, 66336669902188263694667389801400/5006441919390911)
(74256137908911078385783/214422711364, 19967201536746205201236213470948471/99290151878791112)
(3286655922888759117568879/1895817810769, 5955101541900568041861105162047644792/2610326898016296103)
(4070933831527754263670239/111496577929636, 3725895820108072447983693417877834747/1177313996695144873384)
(503636920077693460915759/157659025969, 357358802863141192781337276169942495/62600565828329047)
(7958572770225626610094639/237123226225636, 11055332696893984366956327542494292593/3651414558742680990616)
(38473923605771636193389599/158488510565284, 232395013514901334690210142822584392817/1995247043955705769048)
(37385397070534564485573823/362490489307489, 199074062494781213883649983494938078936/6901522761684826341487)
(7193259033181121575530254911/188201320317141121, 275303476938277036213401430769072748586184/81645828202507239597734081)
(-593228317501404222034254137/134758271346375769, 495305379556452308409733362618251262711371/49469001778878965279690147)
(537498962661602782932574471/16511345629383361, 6465444432869797285232496295822980272560/2121649809950492088452641)
(2723080668102991219229677639/33641473216129, 142098869825951358434924992352625851612087/195124817120646648383)
(10642730097738642227049610351/13873319652493249, 1094872324144420507880840006961567707210489/1634069746048137525580193)
(8470393355410545151355354959/383214139278298084, 1089382892011881926959026911290423859961095/237226000530552617106361448)
(706218495557912502037749678511/19056892466136366241, 267995826202976067885153016883583379796383336/83191341015015520740559153039)
(898057342536905398041918254503/45609590197996256644, 1574294246409180734886442950563731229777629025/308023786427207777168936868728)
(258569207105469643969772815327/383995991450158201, 131006557835883660997102464671541427854395093/237952370789480469512452301)
(230291437484345993354130938551/10225349845269488836, 147292983777970195182323640819598951225633601/32697704414899582601097510584)
(643778450460460495538784167119/7312999924008203881, 428068315634656406118389809891190544330681568/19776223125381737677523312971)

Commentary

From the Layr-Labs elliptic-rank-challenge-public repository, submission/board/1-20-mestre-e-man.ec. The source attributes this curve to its own search using Mestre's ten-root construction (Mestre 1982), with roots [0,1,64,78,101,108,121,137,138,140]: a GPU Mestre–Nagao search followed by point search on the quartic. The displayed model and all 20 rational points are reproduced from that source. All points were checked exactly on this model, which was independently confirmed minimal. The minimal discriminant was fully factored and every bad-reduction prime proved prime. The source claims rank 20; no matching upper-bound certificate is asserted here. Source at commit 140b9d566eeaefe70e423bca75e3c5f091b0eaab: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/140b9d566eeaefe70e423bca75e3c5f091b0eaab/submission/board/1-20-mestre-e-man.ec

last edited by The Wren at 2026-11-01 14:05:10 UTC · history