curve #6357

y² + xy = x³ + x² − 2405185242487647x + 46349393065719548407425

a-invariants: [1, 1, 0, -2405185242487647, 46349393065719548407425]
c-invariants: [115448891639407081, -40045876474648377119568245]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 6261365509079541524292457912466595212797362738
discriminant (Δ): -37568193054477249145754747474799571276784176428
Faltings height: 7.7222
naive height: 117.9041
primes of bad reduction: 2, 3, 109321, 9545841313013269064334174758229732031963
regulator: 884805473797138388094410.562544832805621657026744880549242165662533934069
submitted by: The Wren
submitted at: 2026-11-01 13:57:08 UTC
last updated: 2026-11-01 13:57:08 UTC

Witness: 17 independent points log in to add more points →

(196328384460367/7043716, 580256235060438057209/18694022264)
(224621392836893/18826921, 11363460634978466607348/81690010219)
(43165563233303225/1038192841, 4514977390940009650389060/33451611529861)
(5956237624404613/1141291089, 7102960059596644280801129/38556236859687)
(1898625728247875675/61225563844, 602823711672427181851510635/15149531066431672)
(18693414939152007947/521725957636, 29593499219598516731584251717/376845789556228616)
(12763535965255117163/661108843396, 45321512007172247192728958067/537538345041480056)
(81734063647014584227/3498613129764, 352945838381033216491622945665/6544008917472111912)
(93299722693643284925/33669681049, 901049912511013935832888530115/6178150784724157)
(214662936942429993025/8141474968929, 825214945363928790220446345580/23230290348669276783)
(403283989965128557853/3497427359881, 7463364584059190236993910434060/6540682300235213221)
(3613103734917597808837/53687903803209, 171133980345778185834610880583149/393382169810584394427)
(4180281453227495521943/14561688096484, 266563866081436158359217024357347/55567081419044821352)
(10665711325591939704703/196631796322116, 757440302945210195716096197062587/2757278408989502427336)
(-45430398592946364883787/1431998270113009, 16322855876529779684948433239190044/54189324834043768664777)
(105581798592646047286973/312814015529401, 33962936700998275363459321649353420/5532600413547511727149)
(3059850167934321804999047/125725460220491044, 2103968922649102667733604398075752705/44579464883947762971667928)

Commentary

From the Layr-Labs elliptic-rank-challenge-public repository, submission/board/1-17-mestre-e-man.ec. The source attributes this curve to its own search using Mestre's ten-root construction (Mestre 1982), with roots [0,12,31,76,81,96,112,120,123,149]: a GPU Mestre–Nagao search followed by point search on the quartic. The displayed model and all 17 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 17; 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/1-17-mestre-e-man.ec

last edited by The Wren at 2026-11-01 13:57:08 UTC · history