curve #4578

y² + xy + y = x³ − x² − 5207527493333870592479132253792081772037x + 144620420323125482772754602596133271881701574364254541464061

a-invariants: [1, -1, 1, -5207527493333870592479132253792081772037, 144620420323125482772754602596133271881701574364254541464061]
c-invariants: [249961319680025788438998348182019925057761, -124952043159180417114535150704499030857814667683896834162189009]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 483328727128257581098535465545997674973027785591045803016636754582035590
discriminant (Δ): 2740435427455101380021539928002267778666061920104593483274276126046064428701780515269744168344540708205323348224000000 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
Faltings height: 21.5394 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
naive height: 285.9664 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
primes of bad reduction: 2, 3, 5, 7, 11, 37, 41, 43, 71, 113, 127, 131, 149, 337, 401, 1039, 1579, 16661, 22961, 32183, 131581, 162947, 164117, 1974221, 2835163
regulator: 335830005340227485701.69921974272339952757245772931859592870143922435826272
submitted by: jesper-petersen
submitted at: 2026-10-18 08:08:39 UTC
last updated: 2026-10-18 08:08:39 UTC

Witness: 18 independent points log in to add more points →

(6358319309605088281, 334314751790829581304985376084)
(17558963446770353161, 242064796724378451770274875924)
(23517000666444077731, 187512810289322774545749259484)
(37213695592889277631, 48629248975055082681326941184)
(38935176922128215191, 29801337270719666119519501064)
(42828689304899740831, 12221440968418976930084002784)
(52920621772315790431, 131316063340518690985767315584)
(69832634848090393591, 348583798108877321788040471264)
(132152956019021184031, 1328308267531781365638369075584)
(-43339503299083339434929/529, 1778105204750596803212751390405088/12167)
(-22042966266325234547369/361, 3324567550881589212800459797447616/6859)
(3053731267630061675269/49, 85496744685963375232591924800812/343)
(35393446809508652028151/841, 14468266905323310132788671919296/24389)
(113150318643740829710389/1369, 26746983044854126569113254368880502/50653)
(272351894660504881440063091/6630625, 78555455327815170373086462590296494164/17073859375)
(691396313439248446073102279/16785409, 167645667086400004300718062278445123232/68769820673)
(389198224543902574791763035481/9232903744, 2539150056478023646520456902151924601858233/887171254953472)
(118173422614763973620489369551/2402862361, 10176786080760171574435941271273323966971176/117785910073859)

Commentary

Curve and independent-point witnesses from the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r18-z2k3.ec. Specialization u = 4/7, t = -12673/12022 of the Elkies–Klagsbrun Z/2Z family (arXiv:2003.00077). The original submission describes Mestre–Nagao sieving followed by PARI/GP rank calculations. Submitted to ICARM to document these improvements and make the curve available to other researchers.

last edited by jesper-petersen at 2026-10-18 08:08:39 UTC · history