curve #182

y² + xy = x³ − 705592481967814336771305934410x + 228068530382732810578825694652792716443320100

a-invariants: [1, 0, 0, -705592481967814336771305934410, 228068530382732810578825694652792716443320100]
rank (lower bound): ≥ 16
conductor (N): 15495342387615096338864350597570025083525109638868085907030
naive height: 217.8001
Faltings height: 15.8767
discriminant (Δ): 11765406596767129399549312051539530956680847347822041839788385072806810895769600000000000
primes of bad reduction: 2, 5, 7, 13, 17, 19, 23, 31, 157, 2520531720283, 36758332594990127793731213603179
regulator: 398172387122838.1436246285395286883159166095411598056198651333868740662
submitted by: Seewoo Lee
last updated: 2026-07-01 22:41:37

Witness: 16 independent points

(85806452822380243564820/151321, 25135032253556314427798345617442510/58863869)
(143543240122282594/9, 54310424664781703112741884/27)
(21932292098978439904820/3568321, 3219209126882113291254038149097510/6740558369)
(130893338118852930/361, -30597161802530398523394340/6859)
(414053988634420, -2626948398704820269210)
(478311460656900, -67395889520080308010)
(20822298416231551497220/43943641, 100561680095391510410886047503310/291302396189)
(995281150564688360980/1907161, -3697510664774461251644234425970/2633789341)
(474209115686009700/841, -75118812688062079506219410/24389)
(775816978888420, -12149729763811122725210)
(23783724072221805/16, -3176744871249302298563455/64)
(7460889361050978666716020/26224641, -20379053492046522649000488415619494810/134296386561)
(1753748568441128308/1089, 2054491546835926432213605734/35937)
(4980692083756696180/7569, -4595470418622623132327656630/658503)
(467596474010420, -611791578487052205210)
(3097490471898420, -166616961664972326125210)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=389/10: 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 16:08:57 · history