curve #163

y² + xy = x³ − 22636258359102379085224610x + 41564547766350787714309803970639244100

a-invariants: [1, 0, 0, -22636258359102379085224610, 41564547766350787714309803970639244100]
rank (lower bound): ≥ 17
conductor (N): 253719162614603331010718284712814989331353661834356557470
naive height: 186.7638
Faltings height: 13.3744
discriminant (Δ): -4003543597440842456820042201225551911119601184817241350745392322788249600000
primes of bad reduction: 2, 3, 5, 7, 53, 113, 733, 161647205569, 98509003985351, 326103671223493957373
regulator: 16709644970985311.2146576317744156672737110289849683945374146824957952763
submitted by: Seewoo Lee
last updated: 2026-07-01 22:36:39

Witness: 17 independent points

(41963813756980, -270163542135951662210)
(36238334888500, -216356341349008698290)
(182911406772646514980/24433249, -2062545615877669218157217269870/120773549807)
(3761563658380, -3104924371751178170)
(98561136705675460/26569, -12731169053892275162899910/4330747)
(5138189206478380/1849, -27554611597486805688470/79507)
(66765098900596/25, -49867050842311805146/125)
(2338487488420, -1190805577977135410)
(1512239740180, -3285026256352586210)
(479603888121063940/94249, 220635347202511675210409050/28934443)
(372914677744663396/9025, 226279698064107559786550834/857375)
(84032515043620, 769111086316879443790)
(363489079868260, 6929470060065718088590)
(421895389051189638601540/64821669201, -215147281405435544470104279938143810/16503661800243801)
(9783488506214227063300/49042009, -967427023992792791305830551022230/343441189027)
(120624113779620, -1323787532231407812210)
(466217485868799160/289, -318332482341864215809401190/4913)

Commentary

Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-324, 24, 120, 180, 276, -276] with shift t=1355/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:07:17 · history