curve #167

y² + xy + y = x³ − x² − 4322087140978044344318x + 113218873181275746647084844047757

a-invariants: [1, -1, 1, -4322087140978044344318, 113218873181275746647084844047757]
rank (lower bound): ≥ 16
conductor (N): 56059062740283999533055887298056005020483418801410
naive height: 161.1369
Faltings height: 11.3485
discriminant (Δ): -370335152258882249448312295653484868140997279261599897092748492800
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 43, 6529, 1327973, 2253551, 11863534244867284689946519
regulator: 129652450970271.3285372382617348758143858676004025284640855566598266686
submitted by: Seewoo Lee
last updated: 2026-07-01 22:37:41

Witness: 16 independent points

(4932628938917, -10954167952365331425)
(446184178743003/196, -9420897930023067416333/2744)
(9544497553624449113/5909761, -29462901095704390278748461027/14366628991)
(230856486055446737/259081, -110627478320435505484822737/131872229)
(687244594339217/1369, -17869337246424980997321/50653)
(111714407682185/358801, 2273230596607997710939317/214921799)
(20777055868853/841, 113138478760251712407/24389)
(4297326630785/121, 2828314365172462761/1331)
(38392883723, 1968003627921723)
(43458840423, 2732270569874853)
(205418185807217/3481, 1638702900632091787593/205379)
(1009339707253977/11881, 24543070374389616153303/1295029)
(50577086539596065/434281, 9870335059605680578829697/286191179)
(257718003905, 126951044297053971)
(686682213293, 566513798751435243)
(50399148128803/36, 357407945712747949543/216)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1929/1: 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 16:07:44 · history