curve #187

y² + xy = x³ − 1189299700325816602093307071x + 15790705709481148712666898533494358863001

a-invariants: [1, 0, 0, -1189299700325816602093307071, 15790705709481148712666898533494358863001]
rank (lower bound): ≥ 16
conductor (N): 6478350825852857868575711796470819320010194843359558272176170
naive height: 198.6436
Faltings height: 14.2811
discriminant (Δ): -57756388763308488291464979237750795586612877872758868116877248909627520480051200
primes of bad reduction: 2, 3, 5, 7, 13, 23, 31, 43, 3277612446733284530993, 1816531268684434524436436201159
regulator: 6104323411601247.835938464639324428147234046756706737038186530028901018
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:40

Witness: 16 independent points

(697576442233977388406/177241, -18423455744863744566579326547943/74618461)
(1617980147791814, -65067171076314760848307)
(8996486721959734/9, -852813860149243222627361/27)
(2089299406246950440/18769, -2889502216708937627799054803/2571353)
(4959252092193526/361, 310190738443800799859023/6859)
(810579905488502486/52441, 398415122436772395438033593/12008989)
(27510586103866689230/1530169, 28071482427182809558959004193/1892819053)
(18427107540326, 11507788246080029357)
(3346688353809670/169, 4876157311998696125137/2197)
(184998133634752/9, 146224271703375470585/27)
(23177380051814, 26010241714416410093)
(39043352706710, 169922010875689734941)
(59337157252816, 392608088389820811787)
(30436825206632108486/398161, 153149547828250401696053592347/251239591)
(17545171071416, 18033907841991859187)
(19600296319046, 3155410596522222317)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=842/9: 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 16:09:21 · history