curve #166

y² + xy + y = x³ − x² − 3687257198396459595596432x + 2724740878574131022947561407482810739

a-invariants: [1, -1, 1, -3687257198396459595596432, 2724740878574131022947561407482810739]
rank (lower bound): ≥ 17
conductor (N): 283351194095753440937593600911468959073927371811953815851270
naive height: 181.3144
Faltings height: 12.8239
discriminant (Δ): 1153068932160301343602378339411351240373132290030473551029141204930560000
primes of bad reduction: 2, 3, 5, 13, 17, 37, 612349853, 628765815428201506421019641699709465077893163
regulator: 58276555418819039.9906295005332111538296892176393442258933331924181239547
submitted by: Seewoo Lee
last updated: 2026-07-01 22:35:42

Witness: 17 independent points

(23495348691869, -113517683577582019887)
(2761907455655033/289, -142419822038161029626487/4913)
(8514920884797, -24262997061118929199)
(8105002919171097/2401, -630759937393946987000151/117649)
(57797791296753/49, -44176752064533223057/343)
(1154713200669, -81656470446780783)
(4175670052134923/3721, -2520356457515651095617/226981)
(7696384561642703/6724, 34258403517295224331743/551368)
(1197083102477, 161916663098376321)
(29846231699951433/24649, 727314072799652611618793/3869893)
(2348554157956317/1681, 37768070556822972286521/68921)
(1973500432557, 1770348455310631841)
(3999708899746847, -252954569974946715806049)
(79104131395782773/51529, 9660347883013118928105483/11697083)
(35416531938889181/24025, -2613839719916380864518069/3723875)
(6655658584559163/5929, 5725043071098605862733/456533)
(5297203954988124077/2193361, 9133820904479001092726987241/3248367641)

Commentary

Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=3303/2: 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 17 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 16:07:39 · history