curve #184

y² + xy + y = x³ + x² − 1169167431674669428559431663x + 15401506519118600679103992536133903673781

a-invariants: [1, 1, 1, -1169167431674669428559431663, 15401506519118600679103992536133903673781]
rank (lower bound): ≥ 16
conductor (N): 27848933853647946223887115529474676067564332131897391119550
naive height: 198.5937
Faltings height: 14.3199
discriminant (Δ): -188601378345374742510097865390040902199163746431220797536602817961331170304000000
primes of bad reduction: 2, 3, 5, 7, 11, 19, 29, 43, 73, 127, 10976906220863655801963668324973584199589113587
regulator: 478580258045246.6958029338668409040983047906002491042699010978353959955
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:40

Witness: 16 independent points

(1963577490193219370285/100741369, -4274287134927945750647758597824/1011141120653)
(19725482435915, -3767596628797241358)
(44146965146747074355/2217121, -13166106608227591906765664382/3301293169)
(84992020769165771/4225, -1304507955376817204212926/274625)
(43475756338146835/1681, -3420442022630210280753678/68921)
(5169927390880715/121, -277434016070257283551698/1331)
(1045726115326939, -33798511720116140751662)
(15690799565930621520315/8473921, 1965142616992608074707764709942702/24667584031)
(295490081647955, 5046826565144989978122)
(1957220904751499/25, 79422926794702239882258/125)
(134051813545315/9, -970332229292678234186/27)
(23850116890041395/1369, -906698356527892497902934/50653)
(17622325019835, -16450132122712468718)
(18011746641515, -13644940173569852958)
(2649576959335347595/88209, 2251205537386863546278362306/26198073)
(38852091226715, -169184993480190893358)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=2919/8: 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 16 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 16:09:07 · history