curve #4307

y² = x³ + x² − 163024278446467966212077946127443346832783065x + 726131126127219325552212857544625686433616826753096873828175874663

a-invariants: [0, 1, 0, -163024278446467966212077946127443346832783065, 726131126127219325552212857544625686433616826753096873828175874663]
c-invariants: [7825165365430462378179741414117280647973587136, -627377292973917497277158859910749175852914016763160402671431797231616]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2537399185778233965577407252996697760246054903906475840
discriminant (Δ): 49512586697712116655104697187703091275462385607012996057289139952436611028371529428149347266668321002765669520859436584270440000000000
Faltings height: 24.3890
naive height: 317.0210
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 43, 53, 109, 127, 149, 157, 167, 257, 283, 431, 463, 1039, 2287, 2381, 6577, 6733
regulator: 1955416789967965.338684659820298613160813213369137527099908204100
submitted by: jesper-petersen
submitted at: 2026-10-17 09:25:31 UTC
last updated: 2026-10-17 09:25:31 UTC

Witness: 13 independent points log in to add more points →

(433465557457099827772639/9, 276134624340372855122269186406803600/27)
(12021091341330872800007, -709597679895746922159188637625440)
(650928946751838417671, 787584730925268836165568285180000)
(571022782482037387299791/121, -331116506245124544697479292295620800/1331)
(8354425847651312464581721/576, 16431815514512240703704510451368508925/13824)
(189385992412294532698361/16, -43163351973979579818780683983183125/64)
(65162600397060141229391/25, -70590801603406388853663254267658864/125)
(9303436241042844164831, 121219127198414387515048578509880)
(44258722097190758255546, 8955802356583062465084425954176125)
(277882953456356497463651, 146332670350872264590114028507216240)
(-919991611247484961477228017361/159946609, 2455512312831459838891811171284786129868535000/2022844764023)
(-318826459328290287860736229/58081, 16887865331562829945418952944820227022840/13997521)
(1556084027875685429805994/289, 327680691331785400659387428185562325/4913)

Commentary

Curve and 13 rational points reproduced from the Layr Labs / Yukon Elliptic Rank Challenge (October 2026), file `submission/board/2x2-13.ec`. Original discovery and credit belong to the Yukon challenge participant who constructed this curve; the ICARM submitter is not the discoverer. Source: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/495b451204d7023ddb5e459a89e63df75a78924b/submission/board/2x2-13.ec Torsion Z/2Z x Z/2Z, rank lower bound 13 (13 supplied rational points). Naive height approximately 317.0210, improving on the October 15 ICARM snapshot's previous minimum of approximately 336.9906 for this torsion group and rank threshold.

last edited by jesper-petersen at 2026-10-17 09:25:31 UTC · history