curve #180

y² + xy = x³ − 196802191633137450791741x + 32257087916682386944199682931390321

a-invariants: [1, 0, 0, -196802191633137450791741, 32257087916682386944199682931390321]
rank (lower bound): ≥ 16
conductor (N): 118450782829775208067766915296945483495715377841216329370
naive height: 172.5231
Faltings height: 12.3057
discriminant (Δ): 38326897051202141276118593252174781295683024000118658651028382352998400
primes of bad reduction: 2, 5, 11, 13, 17, 19, 23, 773, 429409, 1975931940579152131201026051304307748559
regulator: 2729177115059906.931362570259210191760555095033069027438437708347161738
submitted by: Seewoo Lee
last updated: 2026-07-01 22:39:38

Witness: 16 independent points

(960455977590754/9, 29765448306343424415989/27)
(1184996775243730/81, 40773516619417655092631/729)
(93926389235815/36, 897876538745062245611/216)
(1581689387006386/841, 61286211837482843133963/24389)
(969908831278, 868211808480707547)
(4642405601764/9, 7040144543520120709/27)
(83217564981347554/263169, 5474064368994652222878799/135005697)
(502451208525362754/1682209, -24075177825653767340300449/2181825073)
(36260210765869890/117649, -1195938362197546351593943/40353607)
(45170047429741794/134689, -3116501815532706470874079/49430863)
(396119718018, -128278541487042673)
(412971536906, -146334814869525793)
(2289306880424454322/6235009, 1517199586293175196745070511/15568817473)
(969779882324404774/2712609, -389292386971292995061885239/4467667023)
(60899311327155256386/19351201, 470748801555341867901936180193/85125933199)
(146178188707299250/175561, -49106110071703841753789451/73560059)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2594/1: 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 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 16:08:47 · history