curve #178

y² + xy + y = x³ − x² − 231933522541838620953102503x + 1359827717462320292062961439163457381087

a-invariants: [1, -1, 1, -231933522541838620953102503, 1359827717462320292062961439163457381087]
rank (lower bound): ≥ 16
conductor (N): 29000334883318273334379766369449958734968994052897662690
naive height: 193.7395
Faltings height: 13.8641
discriminant (Δ): -332816996738660684218455986579305397186353893376619101695289646827493750937600
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 3294355577207735591, 1011771035177430581876701163
regulator: 1597599608068227.671771351581259324452130616281157992166321160531013830
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:38

Witness: 16 independent points

(39311985292419612185763/143592289, 7782675174704441356447161385367794/1720666399087)
(105538367117583, 1073501176971972464158)
(91326967639593, 861336775954602458668)
(6271796033530083/289, 397534517028079176885710/4913)
(990291502852785/64, 19668508858275895940171/512)
(115917997431619887/10609, 12463248179136640291539886/1092727)
(1489574132659398015195/209989081, -25735861290825681984151049464694/3042951772771)
(1805829793061414733123/213773641, -5752256085260123832006215471362/3125584405061)
(2185511165043919158579/247275625, -2262155248862281805639449793842/3888409203125)
(2476891705164386445285/274200481, -6129616608644242348739175367258/4540485764879)
(56792205463555933683/5958481, -56477693860551678057262392322/14544652121)
(187821875570497/9, -2022024040126227255884/27)
(63655414911651003319473/2863213081, -12992980561964414368856652135329918/153207668751229)
(37679089417444755/961, -6828593700442511025350654/29791)
(1147562992021579/25, -36961131093176466599818/125)
(3504789305602515987/3721, -6560492695982148303386152730/226981)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=1851/11: 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 16:08:38 · history