curve #191

y² + xy = x³ − 60923262881679161131199816568720x + 182435750535622934108717118205361600171372294400

a-invariants: [1, 0, 0, -60923262881679161131199816568720, 182435750535622934108717118205361600171372294400]
rank (lower bound): ≥ 16
conductor (N): 28964385051890469693884372342725226899763554916835745467259598596770
naive height: 231.1751
Faltings height: 17.0830
discriminant (Δ): 93858557525002069986716442228582690235767407598682422890314976013836608666063661484178636800000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 73, 5413, 33361703, 2721505543331, 57333637865117140073296578053387
regulator: 4654554321845412.897032645881460013318440912154139638548237775729768145
submitted by: Seewoo Lee
last updated: 2026-07-01 22:41:38

Witness: 16 independent points

(948120211437022752, -923167350663906647619020688)
(21435739948266173578080/67081, -3137468248676342929871311536556080/17373979)
(243859843230968400, -120362332081758768476537280)
(104222286079490939040/5041, -997050253259905842053863275120/357911)
(2739289250202526800/169, -4082763666617324166773111760/2197)
(143889931307632230060/37249, 494799577777326710719500807360/7189057)
(4294930541891040, 834638007028753640880)
(107349527602354836/25, -306438416685685033765056/125)
(6556285940498177654880/1560001, -48678867191471538240064329846480/1948441249)
(104506446187837900/9, 27543714424019930053480540/27)
(12320874007946940, 1141124473343389305468480)
(13284156751610400, 1310481222233810387940720)
(103260000815504640, 33089491507442908457627280)
(1109432993238331471530, -36953162831912838804179582295510)
(5105061389726048340/1849, 14932639441279648341391124160/79507)
(12386226484516320, 1152433950992307113428080)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1295/6: 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 16:09:40 · history