curve #181

y² + xy = x³ − 3994635920327174224332x + 100608973921885468649536289354896

a-invariants: [1, 0, 0, -3994635920327174224332, 100608973921885468649536289354896]
rank (lower bound): ≥ 16
conductor (N): 2612385923234423880735212879508736416654664660133355599474
naive height: 160.9007
Faltings height: 11.3290
discriminant (Δ): -293231918449934958705327201280235162673170146698422177169348591616
primes of bad reduction: 2, 13, 17, 31, 11959, 23251, 408088463, 1680206082665728718324081958011335561
regulator: 18179029818469538.42785326452085272304280545004754511587406136711446082
submitted by: Seewoo Lee
last updated: 2026-07-01 22:37:40

Witness: 16 independent points

(87115448112040/81, 811725076551771850436/729)
(54853647379656152/78961, 12795840505903551940628820/22188041)
(684286234212, 563721713419195768)
(2077475676535/36, 1704402380888340179/216)
(49957756120, 5072387202031156)
(46305881816, 3863202607195380)
(25675060780, 3869312490662116)
(19844421339720/361, -47088243196964595284/6859)
(77021844495756/625, -601082613854102469296/15625)
(176685692356, -70075466002658780)
(391743880410598552/1771561, -236149425639728063915026180/2357947691)
(33693387294, 2065506386879344)
(234291431845531154/6446521, 30331796260318243907203456/16367716819)
(20977572027526184/5329, -3037926865430113221598380/389017)
(573999989640, -432350725599294716)
(27445975192, 3412733323302388)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=281/2: 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 16:08:52 · history