curve #195

y² + xy + y = x³ − x² − 54431691335588366573x + 180548233661579697703929158997

a-invariants: [1, -1, 1, -54431691335588366573, 180548233661579697703929158997]
rank (lower bound): ≥ 14
conductor (N): 4372904911177417822247776105333672098390
naive height: 148.2547
Faltings height: 10.3449
discriminant (Δ): -3760865945124333458688931654370706160938454707422168471961600
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 97, 101, 260763461711492729725756564097
regulator: 249748452478.896612615657083847531400376102137891106663850206439465
submitted by: Seewoo Lee
last updated: 2026-07-01 22:44:38

Witness: 14 independent points

(207486257967/49, -55292996762624536/343)
(5474933974027/1089, -6623280826452051464/35937)
(3485922442500407/683929, -106607557059189060962616/565609283)
(1698397246847/289, -1239319603387616376/4913)
(6268503683, -292677744446952)
(6886225535, -363685789290072)
(227040190559/25, -82465709962026408/125)
(13210232763, -1329222140213832)
(633401232898343/44521, -14202660571046573935992/9393931)
(137061348627, -50670858651120040)
(322490875343, 183089421925893528)
(85458871823, 24892801911837528)
(19817742035, 2623939877550036)
(18737544623, 2395674470375448)

Commentary

Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=491/4: 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 14 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 16:10:00 · history