curve #179

y² + xy + y = x³ + x² − 5156525671696520871642x + 143887580422027574095483236694887

a-invariants: [1, 1, 1, -5156525671696520871642, 143887580422027574095483236694887]
rank (lower bound): ≥ 16
conductor (N): 33158290302303235820773702112187496615798899795769945854
naive height: 161.6163
Faltings height: 11.3308
discriminant (Δ): -168881732015251142870523473594217561643594320461409532628190621696
primes of bad reduction: 2, 3, 11, 13, 17, 31, 97771, 25996012748249, 27882505940647, 1034774277845385713
regulator: 3308054707823128.622854542820352955876197443222107463050585906051669328
submitted by: Seewoo Lee
last updated: 2026-07-01 22:37:41

Witness: 16 independent points

(1796235055657, -2405486875415698453)
(280260196900225/361, -4672441924825390438711/6859)
(33794917963577/49, -195437004131314663299/343)
(276830759494257/1369, -4348434658144756657809/50653)
(-1713074912095/169, 30688521194867715447/2197)
(4976539371434433/190969, 435305469622794811181391/83453453)
(150079661251/4, 14385645730349821/8)
(1209129808574197/24649, 11595110519046294355607/3869893)
(514060728091/9, 161327383722616421/27)
(4645811742356381/27889, 291137569948406847945789/4657463)
(36052656660133/81, 213816965792357845331/729)
(4850146428169, 10680334290818758187)
(176562964409213586197/4115607409, 336723416420365931137503833799/264028562109577)
(1170267928267, -1263655175584965025)
(644082512583421135/96721, -516876990947020027291331609/30080231)
(4995701755998585/5041, 352195399117596328279933/357911)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2577/4: 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 16:08:43 · history