curve #171

y² + xy = x³ − 243033127794890100145570x + 46065726780475015041673462927320900

a-invariants: [1, 0, 0, -243033127794890100145570, 46065726780475015041673462927320900]
rank (lower bound): ≥ 16
conductor (N): 17001080899169472351595692437551130728110858780220870
naive height: 173.1561
Faltings height: 12.2058
discriminant (Δ): 1979571123617220930817947516260302552311906430241986844098656704000000
primes of bad reduction: 2, 5, 17, 19, 31, 1217, 1493, 177263817121, 141808312780621, 3717407278939619
regulator: 503061140161974.4973933639952823682345584297528562360756791803810313595
submitted by: Seewoo Lee
last updated: 2026-07-01 22:39:38

Witness: 16 independent points

(804935286279771142485/2640726544, -2376316883463090509738789991615/135701655643072)
(207118739029424340/436921, -55839103538713816321049370/288804781)
(13398478215906/25, -33013515993024297636/125)
(6136628085887340/10201, -353267915888726695810770/1030301)
(815682344620, -624926083455239770)
(241827032650000350/11881, -118886100875419521952688880/1295029)
(73377184147540, -628538393730579188770)
(337531713230, 49885061341813890)
(965139932699684/3025, 5269925999304849389538/166375)
(744546149483840/2401, 2685159731120074344270/117649)
(1591318012724410480/5294601, 159323126319820860385838630/12182876901)
(24442710167551460/83521, 58393952082645620285070/24137569)
(2499390306144935/7396, -31985792865062494405965/636056)
(20253971560627616260/66601921, -9123104136005428826183866290/543538277281)
(2642069083118124063920/8657744209, -14395846477111130735943458783230/805577125414823)
(89533067998948/9, -846158145396445374118/27)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1355/2: 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 16:08:04 · history