curve #169

y² = x³ + x² − 12539168820487782745280x + 540409134152049294268977949499028

a-invariants: [0, 1, 0, -12539168820487782745280, 540409134152049294268977949499028]
rank (lower bound): ≥ 16
conductor (N): 2411607429845182224131866326159229163946583369283280
naive height: 164.2630
Faltings height: 11.3713
discriminant (Δ): 16592587422778666947058928147605996735160005448846489950560000000
primes of bad reduction: 2, 3, 5, 11, 13, 397, 2467691, 11209488223933, 6398707603623935148158719
regulator: 165765328977244.6178705897496116157230940079152440760508150581557650129
submitted by: Seewoo Lee
last updated: 2026-07-01 22:38:39

Witness: 16 independent points

(647952695791, 514250743863915630)
(28807123943701756/67081, 4737436060388057145083250/17373979)
(402262930006, 246147779511433260)
(276874758276076/1521, 3885218349856161135290/59319)
(6505232023429951/579121, -8824954679706139266996990/440711081)
(11163916115709526/219961, -608011399464814507106580/103161709)
(57498275851, -3085582681209600)
(750610349688556/11881, -800744222870078099250/1295029)
(64222059796, 5807235473910)
(34850173328686/361, -103930279989431903700/6859)
(407905078926, -251586046292995500)
(841209759076, -765023428267291050)
(77470115910364/9, -681812157670903246130/27)
(3572373541773499774/196249, -6751917052827679352774129700/86938307)
(14946749973459694/625, -1827323343746079478328628/15625)
(74724941345461609/73984, 20306169030417705203904075/20123648)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2892/5: 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 16:07:54 · history