curve #192

y² + xy = x³ − 851556499645504323141x + 9487026023034811912831219901121

a-invariants: [1, 0, 0, -851556499645504323141, 9487026023034811912831219901121]
rank (lower bound): ≥ 15
conductor (N): 562577168211394579270024943629595507978443710
naive height: 156.1944
Faltings height: 10.8724
discriminant (Δ): 638730941704979414939006251941574649893567559880993486621081600
primes of bad reduction: 2, 3, 5, 11, 19, 103, 109, 6211, 64376373383029, 19987699093727394221
regulator: 6081709767502.5731747914878712722800396843622123174768992444677519175
submitted by: Seewoo Lee
last updated: 2026-07-01 22:42:38

Witness: 15 independent points

(84305679874/9, -41233185475744451/27)
(36241602249954/13225, -4073566388142236145363/1520875)
(-14513645610267290/2047761, -11411968604968034879033251/2930345991)
(5854928580399234/1075369, -2496576161553282539156589/1115157653)
(2941002666, -2647270682374113)
(600692104935810/326041, -524072430071294943344139/186169411)
(-7819076214, -3958198824213513)
(-23145631818534/1681, -297253625339500924833/68921)
(7156924540410/49, 18790187467681606353/343)
(2536220959236, -4038798111452968923)
(124435835730, -42782323227585561)
(41205269280306/961, -214420931827589747943/29791)
(21031135188, -938135835012105)
(42768585876114/2809, -33673699761100504821/148877)
(5156498703514/441, -9838663942889014997/9261)

Commentary

Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-125, -99, -26, -18, 125, 143] with shift t=593/6: 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 15 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 16:09:45 · history