curve #183

y² + xy + y = x³ + x² − 323202197541587323322357413x + 2206561434845879634664264564474894007531

a-invariants: [1, 1, 1, -323202197541587323322357413, 2206561434845879634664264564474894007531]
rank (lower bound): ≥ 16
conductor (N): 26383630742301813121429708006989650772225567353794271438550
naive height: 194.7346
Faltings height: 14.1058
discriminant (Δ): 57371302902497504412701897991058097709584123040592403193410870194441105536000000
primes of bad reduction: 2, 3, 5, 7, 11, 37, 191, 199, 727, 971, 2300968998537812671658704438428873760173181
regulator: 3263598764701149.083111521840587555373665639651653710373481275686837503
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:39

Witness: 16 independent points

(180924565545781/9, 1671651863450906696876/27)
(10284908500556285015/410881, 26073080629552408556365503132/263374721)
(40246669791815, 233216790682242992892)
(514132005303891/4, 11549195090794268836181/8)
(43617102608779015/289, 9047390518527546741124316/4913)
(6630479150425740485/961, 17073238029898484032493832762/29791)
(982353939816655, -30784333888025912219828)
(366922387740479, -7020194305998509901012)
(5076217626798439755/44521, -11302507691667110491785987368/9393931)
(4254910515588735/361, -40307285259491291477612/6859)
(1036183133024957815/89401, -111930322750105636756280092/26730899)
(2518861545901311335/214369, 554226264746709541557734724/99252847)
(305995178478481/25, 1147752566954104940114/125)
(12454361849885, 10634580957552665982)
(6677202029640505315/316969, 12294351622120792432232250204/178453547)
(247425027171655, 3881930663163099914172)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=3303/11: 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 16:09:02 · history