curve #185

y² + xy = x³ − 13926389068853423234936756x + 18667175679332355979688921260281429136

a-invariants: [1, 0, 0, -13926389068853423234936756, 18667175679332355979688921260281429136]
rank (lower bound): ≥ 16
conductor (N): 43657243228043341071443078601634748045721626333578493820310
naive height: 185.3011
Faltings height: 13.3955
discriminant (Δ): 22324203557420732705257905143620661610548249883929708008721170067485403955200
primes of bad reduction: 2, 5, 7, 11, 23, 29, 53, 67, 127, 2749, 158647, 266621837, 124179295151, 13053706547096656897
regulator: 836077628215322.2135091501443589696508094598206988819231525455259700725
submitted by: Seewoo Lee
last updated: 2026-07-01 22:39:40

Witness: 16 independent points

(152286804214184, -1878728999700747104692)
(765300447413032/9, -21151245578527565284156/27)
(281550663104079/4, -4717753822362220088699/8)
(486836121081064/9, -10716771708385511474876/27)
(43233200766510, -283238750412797198806)
(1184091393673305064/1071225, 2384138221407547739830063268/1108717875)
(2991492522774306/2209, 157154578672959972375304/103823)
(221345423961929704/133225, 16527671876532706844540028/48627125)
(3240266480360533454/1934881, -550089703964649232827177002/2691419471)
(136199984927144/25, 1277794623832474021468/125)
(11198548828296616/1681, 1025845985070022440673452/68921)
(20410091612901109370/2337841, 84777817445557080735804895966/3574558889)
(3669418539293138073664/25170289, 222205513477376556361838615217876/126279339913)
(209617558513521864/124609, 958837106568577566899084/43986977)
(2587523431659102184/16129, 4161116451083341379626226036/2048383)
(1284015905840, 1703649433200821444)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=89/4: 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 16:09:11 · history