curve #6418

y² + xy + y = x³ − x² − 1047550570326311382845142x + 413876038132509334479477194584200165

a-invariants: [1, -1, 1, -1047550570326311382845142, 413876038132509334479477194584200165]
c-invariants: [50282427375662946376566801, -357588896946261794067077812862054392185]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 1685454917805383803661722878638558094370688613417272529519797690854
discriminant (Δ): -428024647286512827707501848476019105181625114754543193416611663187755008 ★ record for rank ≥ 21
Faltings height: 12.6092
naive height: 177.5449
primes of bad reduction: 2, 31, 22067313251, 1231901520116619155808160457660558454281984448893045167
regulator: 90675234578361487613.777284126280685798903038254748702765823809036231623463489358
submitted by: apkorte
submitted at: 2026-11-02 14:15:54 UTC
last updated: 2026-11-02 14:15:54 UTC

Witness: 21 independent points log in to add more points →

(607343527465, -41005846515648501)
(429327934089, 208009648836964395)
(752612212313, -227541944402357893)
(578241609973, -38490770207801753)
(694007634997, 145376268274700835)
(492010837393, 132563561332852027)
(603860568277, 38701815459769927)
(1248920122089, 1026467346783581195)
(623152623867, 55445836953178307)
(5853988706029/9, -2367868728738241843/27)
(17404057026849/25, 18540566137872292383/125)
(698018126095, 150869257105586379)
(2019564974027/4, 935736887201955181/8)
(5334984676369/9, 937183690750504297/27)
(296716735075, -359407212844019991)
(854640678451, -377936129545337069)
(2994078089663/4, 1773501702287171911/8)
(48118068420313/81, 25422326819799931891/729)
(31230273433321, -174434875974044015093)
(23011399708601/25, 59882413580778741559/125)
(85916533488301/9, 791970452952867528205/27)

Commentary

From the Layr-Labs elliptic-rank-challenge-public repository (https://github.com/Layr-Labs/elliptic-rank-challenge-public, file submission/board/1-21-m10-131401.ec), where it is recorded as a find of that challenge; not our discovery. Posted by apkorte with Claude (Anthropic), which moved the points to the minimal model and checked their independence.

last edited by apkorte at 2026-11-02 14:15:54 UTC · history