curve #189

y² + xy = x³ − 165634403178213270286613816x + 799855393883091839227487570305634706496

a-invariants: [1, 0, 0, -165634403178213270286613816, 799855393883091839227487570305634706496]
rank (lower bound): ≥ 17
conductor (N): 8541964435847820105889019133681263942251915526797299744486693070
naive height: 192.7291
Faltings height: 13.9675
discriminant (Δ): 14444861515529646982364107927886725142451411941867709623555198879792460966963200
primes of bad reduction: 2, 5, 7, 11, 13, 31, 963617201, 156542453132674409, 182484217080500391795285792738533
regulator: 174183243424871196.117456918605717418856645255314545313199949862786374577
submitted by: Seewoo Lee
last updated: 2026-07-01 22:36:40

Witness: 17 independent points

(3771724805693216/25, -230820834416887219151576/125)
(220650513021776144/2209, -102825180245912272434259304/103823)
(58363265298704, -435814179298469087992)
(121206730250907905696/4490161, -1203454835017313260439695009672/9514651159)
(22736561402976, -93742167691902387368)
(89987241840089645664/5678689, -628103463007840626180227153096/13532315887)
(564525651073621876/63001, 93736751901381586320629692/15813251)
(1449414454105204150/151321, 564974171284448928893290084/58863869)
(273661363709014/25, 2159293870027528337902/125)
(13964147866804, 34783393964917112948)
(833419877491021060/56169, 533954318650346565473505592/13312053)
(11612977777536, -21035392256727489208)
(3226174044596208007696/47045881, 180219935774189259169905542093208/322687697779)
(573546865748084535976/11215801, -13336036910237094494990068561432/37561717549)
(43838486531548, -278905136311491012660)
(5538299763932867296/477481, 6914201219537011087158124872/329939371)
(12688301446408259420456/1434818641, -280680749199495086182087317126472/54349495302439)

Commentary

Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=3023/1: 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 17 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 16:09:31 · history