curve #186

y² + xy + y = x³ − x² − 433375986345644467803047x + 107998191356151184499232541284273119

a-invariants: [1, -1, 1, -433375986345644467803047, 107998191356151184499232541284273119]
rank (lower bound): ≥ 16
conductor (N): 80641417935190941169492920447777598088517349263967556694390
naive height: 174.8913
Faltings height: 12.4617
discriminant (Δ): 170562405115020174321241730237321859944276566942438388072716846080000000
primes of bad reduction: 2, 3, 5, 17, 13219, 32259473509, 8349520517389, 14802985393449629740213309277
regulator: 139758098201894122.2826035804257146048295377794883418723209203275293760
submitted by: Seewoo Lee
last updated: 2026-07-01 22:39:38

Witness: 16 independent points

(117823316540065899/105625, 34543457264653001082310582/34328125)
(3714470393587, 7053214591294192926)
(180092834680363333/21904, 76188679591183554465639687/3241792)
(10159084638447, 32313984927375252466)
(413500337987667, 8408397657985911069886)
(8822791558227, -26135518036715157314)
(1290399889827, -1302865125960336914)
(796410792327, -517680291105402914)
(156319180011123/289, -876615168560382378082/4913)
(1577789313211767/3721, -4942957593625439902634/226981)
(4460191705964563/10609, -11374202741656054438878/1092727)
(16971071132626147/40401, 70123120568310244525166/8120601)
(477541992066122787/1018081, 91420772927296554619724494/1027243729)
(481359300087, 104512157193136426)
(144856432632003/289, 634955008440682893518/4913)
(441065116592403/529, 6949883465755085952962/12167)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2901/10: 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 16:09:17 · history