Rational Points of Projective Curves: X/Y+Y/Z+Z/X=n (n \in [-1000..-501])

Rational Points of Projective Curves: X/Y+Y/Z+Z/X = n (n \in [-1000..-501])

[2010.10.20]X/Y+Y/Z+Z/X=n(n \in [-1000..-501])の有理点

x/y+y/z+z/x=nの有理点を計算するプログラム(pari/gp)

■整数n(n!=3)を固定したとき、射影曲線C_n: X²Z+Y²X+Z²Y=nXYZは、楕円曲線
E_n: Y²+nXY+Y = X³
にQ-isomorphicである。

■n=-501,-502,...,-1000に対して、楕円曲線E_nのねじれ点群E_n(Q)_torsを計算すると、
E_n(Q)_tors =Z/3Z if n=-501,-502,...,-1000
となる。
これより、楕円曲線E_nのねじれ点は、[0,0],[0,-1],Oの３個である。

[pari/gpによる計算]

gp> forstep(i=-501,-1000,-1,print("E_{",i,"}(Q)_{tors}=",elltors(ec(i),1)))
E_{-501}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-502}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-503}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-504}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-505}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-506}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-507}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-508}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-509}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-510}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-511}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-512}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-513}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-514}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-515}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-516}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-517}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-518}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-519}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-520}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-521}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-522}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-523}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-524}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-525}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-526}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-527}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-528}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-529}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-530}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-531}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-532}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-533}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-534}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-535}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-536}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-537}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-538}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-539}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-540}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-541}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-542}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-543}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-544}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-545}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-546}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-547}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-548}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-549}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-550}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-551}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-552}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-553}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-554}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-555}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-556}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-557}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-558}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-559}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-560}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-561}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-562}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-563}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-564}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-565}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-566}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-567}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-568}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-569}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-570}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-571}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-572}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-573}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-574}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-575}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-576}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-577}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-578}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-579}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-580}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-581}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-582}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-583}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-584}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-585}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-586}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-587}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-588}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-589}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-590}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-591}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-592}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-593}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-594}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-595}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-596}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-597}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-598}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-599}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-600}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-601}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-602}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-603}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-604}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-605}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-606}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-607}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-608}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-609}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-610}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-611}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-612}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-613}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-614}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-615}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-616}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-617}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-618}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-619}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-620}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-621}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-622}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-623}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-624}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-625}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-626}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-627}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-628}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-629}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-630}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-631}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-632}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-633}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-634}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-635}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-636}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-637}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-638}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-639}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-640}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-641}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-642}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-643}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-644}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-645}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-646}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-647}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-648}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-649}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-650}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-651}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-652}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-653}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-654}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-655}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-656}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-657}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-658}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-659}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-660}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-661}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-662}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-663}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-664}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-665}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-666}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-667}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-668}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-669}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-670}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-671}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-672}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-673}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-674}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-675}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-676}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-677}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-678}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-679}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-680}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-681}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-682}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-683}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-684}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-685}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-686}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-687}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-688}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-689}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-690}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-691}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-692}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-693}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-694}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-695}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-696}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-697}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-698}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-699}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-700}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-701}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-702}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-703}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-704}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-705}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-706}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-707}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-708}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-709}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-710}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-711}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-712}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-713}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-714}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-715}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-716}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-717}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-718}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-719}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-720}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-721}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-722}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-723}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-724}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-725}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-726}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-727}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-728}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-729}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-730}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-731}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-732}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-733}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-734}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-735}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-736}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-737}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-738}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-739}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-740}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-741}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-742}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-743}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-744}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-745}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-746}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-747}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-748}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-749}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-750}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-751}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-752}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-753}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-754}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-755}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-756}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-757}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-758}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-759}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-760}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-761}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-762}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-763}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-764}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-765}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-766}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-767}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-768}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-769}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-770}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-771}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-772}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-773}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-774}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-775}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-776}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-777}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-778}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-779}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-780}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-781}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-782}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-783}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-784}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-785}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-786}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-787}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-788}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-789}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-790}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-791}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-792}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-793}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-794}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-795}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-796}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-797}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-798}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-799}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-800}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-801}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-802}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-803}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-804}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-805}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-806}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-807}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-808}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-809}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-810}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-811}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-812}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-813}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-814}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-815}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-816}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-817}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-818}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-819}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-820}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-821}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-822}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-823}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-824}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-825}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-826}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-827}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-828}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-829}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-830}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-831}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-832}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-833}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-834}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-835}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-836}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-837}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-838}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-839}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-840}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-841}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-842}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-843}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-844}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-845}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-846}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-847}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-848}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-849}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-850}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-851}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-852}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-853}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-854}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-855}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-856}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-857}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-858}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-859}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-860}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-861}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-862}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-863}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-864}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-865}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-866}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-867}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-868}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-869}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-870}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-871}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-872}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-873}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-874}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-875}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-876}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-877}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-878}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-879}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-880}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-881}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-882}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-883}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-884}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-885}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-886}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-887}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-888}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-889}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-890}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-891}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-892}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-893}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-894}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-895}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-896}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-897}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-898}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-899}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-900}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-901}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-902}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-903}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-904}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-905}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-906}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-907}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-908}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-909}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-910}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-911}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-912}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-913}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-914}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-915}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-916}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-917}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-918}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-919}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-920}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-921}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-922}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-923}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-924}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-925}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-926}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-927}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-928}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-929}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-930}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-931}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-932}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-933}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-934}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-935}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-936}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-937}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-938}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-939}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-940}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-941}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-942}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-943}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-944}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-945}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-946}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-947}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-948}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-949}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-950}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-951}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-952}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-953}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-954}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-955}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-956}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-957}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-958}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-959}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-960}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-961}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-962}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-963}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-964}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-965}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-966}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-967}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-968}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-969}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-970}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-971}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-972}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-973}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-974}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-975}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-976}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-977}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-978}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-979}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-980}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-981}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-982}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-983}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-984}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-985}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-986}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-987}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-988}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-989}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-990}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-991}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-992}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-993}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-994}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-995}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-996}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-997}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-998}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-999}(Q)_{tors}=[3, [3], [[0, 0]]]
E_{-1000}(Q)_{tors}=[3, [3], [[0, 0]]]
time = 243 ms.

■n=-501,-502,...,-1000に対して、楕円曲線E_nのMordell-Weil群のrankと生成元および楕円曲線C_nの自明でない有理点を求めると、以下のようになる。

-	E^~_n: Y²Z+nXYZ+YZ²=X³ E_n: y²+nxy+y=x³				C^~_n: X²Z+Y²X+Z²X=nXYZ C_n: x²+y²x+x=nxy
n	[a₁,a₂,a₃,a₄,a₆] Conductor of E_n	rank(E_n)	E_n(Q)/E_n(Q)_tors の生成元 [X:Y:Z]	E_n(Q)/E_n(Q)_tors の生成元の高さ	C^~_nのXYZ!=0である有理点[X:Y:Z]
-501	[-501, 0, 1, 0, 0] 3493098	1	[-2108186398820688285731658 : -1056559338012189912350209453 : 381215061455206007847987]	39.9639741703553	[829990935515983135096948 : -75217566519398138761219827 : 150083999055769098898422], ...
-502	[-502, 0, 1, 0, 0] 126506035	1	[388008014940 : 68538442009024 : 126241588242125]	21.4492108734855	[-1461846204 : -84015026473040 : 475623645525], ...
-503	[-503, 0, 1, 0, 0] 127263554	1	[-237123603854198959001186793792120052041510316380 : -153832000262069797286856657403919572079568176413167 : 34558830031356382398482194233581964054325892985152]	76.5820564743201	[989072189074964504179343307884670453444193575 : -93515615190524552811611962054320110030423742580212 : 144149199469827494179588743274380204870408057680], ...
-504	[-504, 0, 1, 0, 0] 9848007	0	-	-	-
-505	[-505, 0, 1, 0, 0] 64393826	0	-	-	-
-506	[-506, 0, 1, 0, 0] 129554243	0	-	-	-
-507	[-507, 0, 1, 0, 0] 130323870	1	[-657245913324991888041567620320680034480322633009706330692549360554619299030415699532079056068332572582710929886663222081152526613375716974635903596727278535069755496021770597389752464756983801861345404974557697646597736544255764674723569963236086598453032852 : -333261424775644897297075150886188246887605914765399250638078975779634250778083187512066859057800412286108273103456870050996421012192208716817310299400680571298192675711852682598846594357344967817444597863612309989124763459455724795529861478114012405309121760221 : 53631429029562517194714836288946986640658160399314329436023870913633166847482142481897493797262899132593294924350125940401969323371826681460554578681866645206098939218477569506014907996416119876797917104949782356032210422807429657615373789262563082005170689]	398.366722542523	[438108010329540229711366212302829923972328901182685839348537715679448909022032014324193135764159510680274932262289099726844144875399229005107172105857493233891187155273272467861524188449247638193136037763550331733042899063332311987030918516827394521962043984 : -18127162201354083742351480923247291006072593069523759258456861439995106354966421660418369552514052281411604422324283879309780494569959598860790460452915810342478553378642433461368959804363685323188258512352258567083492944193328812352504698555172193656890914649 : 35749721963890263678082955492815927638927001086369208102753012259138505091578672198727265289733786781346570691270687783405318530221288789961447917491388699814475493224658725339432922742468743910761696512558580057890393859143651214427414362891207228737059588], ...
-508	[-508, 0, 1, 0, 0] 131096539	1	[306499462609802110 : 155635964567116318648 : 68464611012127357]	29.4850322889123	[-104354700886121950 : -11836646820305869012 : 23310331256765965], ...
-509	[-509, 0, 1, 0, 0] 515126	1	[102713888143775002302250863340562058103195351000301630455653228795778108968541406995255296930 : 35810293424541885711995163493236561965382732491758297971721260969508994499885824369670812024 : 52246137967599131517202156887124547361809611699363288574354810239265304556456244265727780632625]	143.915365306463	[-2290267710753516373892826018449531890763654205523114373745707186236802907188265845162499294 : -406153297940164205992694285827537427681762784007054330170204263629625159213725517955058646980 : 1164960697732261975816291485488243564635278529478673040723052788783124330366789771993011752975], ...
-510	[-510, 0, 1, 0, 0] 1637667	1	[9464 : 4826809 : 343]	7.60329557838478	[-10816 : -199927 : 392], ...
-511	[-511, 0, 1, 0, 0] 133432858	2	[-309365993313832 : -158312325285919232 : 227126149929197], [268217349993169668 : -2433068073706912803 : 139492077066769611743]	24.9835920624047, 29.8654799132727	[47525731488872 : -17855233401365312 : 34891800162037], [-1988639007648048 : 9381796757392678983 : 1034233563637437748], ...
-512	[-512, 0, 1, 0, 0] 134217755	1	[-349702512869268 : -207990296615249 : 1149705130176]	25.0444057735650	[18807258845755601 : -36775411354476 : 61831989144432], ...
-513	[-513, 0, 1, 0, 0] 67502862	1	[1271368634586752990940141682995855 : 34552725854794218740879711857 : 733315853907793414390162499737734383]	54.0610570602215	[-610269265048758304294820692443825 : -9566474122301895061574968565423 : 351998716216854066869678713055883345], ...
-514	[-514, 0, 1, 0, 0] 135796771	0	-	-	-
-515	[-515, 0, 1, 0, 0] 136590902	1	[-877055468280431548518433890396 : -451964243791212760569448993762017 : 285898760691895518271455907648]	48.6376083049556	[230964153048355521480106095513 : -38797774053292002563780583827988 : 75288698957937287007625098544], ...
-516	[-516, 0, 1, 0, 0] 137388123	0	-	-	-
-517	[-517, 0, 1, 0, 0] 34547110	1	[-21041927857538035496677592656 : -2071474686456500171139623 : 398808818478481325162308186112]	46.7515112869018	[64106393122661940499875463367 : -119611977738850777931114672 : 1215012002286997948950771683584], ...
-518	[-518, 0, 1, 0, 0] 7315361	1	[-153506995134226642259528752137804465889867351592063340867420 : -267903535358873862958188208542943412105935044273107691759 : 169443586156273508454921426779602934624806906174817346136199]	92.1719320824249	[1193750142922665915413586807483653969739822530780612990415600 : -2299644344511422667010739392861315989416920255065403827639 : 1317681353963789159772104908385983262388213683506133685181820], ...
-519	[-519, 0, 1, 0, 0] 15533154	1	[749228460 : -29791 : 40530337875]	17.0828018169183	[-1534664176 : 3301035 : 83019347100], ...
-520	[-520, 0, 1, 0, 0] 140608027	1	[2699918288909734994140 : -475026710201576000 : 1373799432321081656417591]	36.0392340292901	[-755984706250802589860 : 67678875000797855600 : 384667700706661756852009], ...
-521	[-521, 0, 1, 0, 0] 70710394	0	-	-	-
-522	[-522, 0, 1, 0, 0] 28447335	0	-	-	-
-523	[-523, 0, 1, 0, 0] 143055694	0	-	-	-
-524	[-524, 0, 1, 0, 0] 143877851	0	-	-	-
-525	[-525, 0, 1, 0, 0] 18087894	0	-	-	-
-526	[-526, 0, 1, 0, 0] 6327461	1	[259771346031173765231545385176290984667704 : 135874294760648310481295072487175559237715419 : 765601764785953911197609216537141481012021]	66.4962347906889	[-15684407039715213510682524462390996857024 : -24178304564641917825864960965259123872168661 : 46225305033397293064905476217447814197576], ...
-527	[-527, 0, 1, 0, 0] 146363210	1	[-5903563139236153280322659244355721740437630650062829646175173142344260060020624506702510121424233935756927042219608081743417754383246912078145749924099904737141019480534806081818122859545592323333291475322257894036678212681256961503287961242016669082455927928125550969684247044688962616494227625037040 : -3085370446770367043335886192712149475160147415405585286532844868151772463981749373763107817668206295455445851619166504051750367527792974144517289008259985867994483233924674475199764566242625439291470423925524418065359165756977049904407138676224995546591479588149274747543304918415860368077408063821737984 : 2583740002293924344569772937456926702439877279146364692423224625383428178380401720188630251652225589502572408004417198162556785408558056978696062609284247451018868675481249585831974189082553693808673266031991836760050588392935894073372747091390859754329333401558285213913629641292330399092633504125]	464.684448503288	[127143290527035543754767138956036367314566806857582177817107074914370245832991739957815609388629671101732020012225654940687435143614604251794778970013168591367447942575281758261857002685016993846246118755839436371824608901773012297482693375674136839037790931789799555723056199764278919639278777532474864 : -29081790838513885343884176216049663105761948231195238029649845860359824387221716762195098562963856396084796558603241474929899391332011883781940523706231037760902510601777501340057423478267619409812661820623012263188653085093823351410826346742549899759732479990522355448825171292546296192204269894063360 : 55645243052399087507826613462578154502316263317645916881975589553455119004479969473886190852306345356844595965108668561792259651417443452270942636723347467311709691051013939713377511117279945730266904579127808648638251377317957273167483717459895423887998547139412279110030681208566871774138532199725], ...
-528	[-528, 0, 1, 0, 0] 16355331	0	-	-	-
-529	[-529, 0, 1, 0, 0] 74017958	3	[23 : -1 : 12167], [-1871595 : -5000211 : 1331], [23939370 : 12665630691 : 493039]	4.35666773975729, 12.1923964616684, 13.9502401385670	[-1 : 23 : 529], [169294275 : -321651 : 120395], [-39393900 : -429251319 : 811330], ...
-530	[-530, 0, 1, 0, 0] 148877027	1	[-3772704440232 : -1921794040535552 : 359425431]	22.5743895606477	[226463084113448 : -10990248076224 : 21575130759], ...
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-897	[-897, 0, 1, 0, 0] 8019270	0	-	-	-
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-901	[-901, 0, 1, 0, 0] 182858182	0	-	-	-
-902	[-902, 0, 1, 0, 0] 733870835	1	???	?	?
-903	[-903, 0, 1, 0, 0] 736314354	1	[-25050480 : -170953875 : 4096]	14.6522352815123	[4416712755 : -4928400 : 722176], ...
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-907	[-907, 0, 1, 0, 0] 746142670	2	[278251465072 : -697628098909 : 122023936], [-7924213823951994829789477896052 : 8735625093986121050890513424 : 70673278488166002200994828211]	20.7119544387739, 47.7352801195861	[-35484354677821 : 39014943856 : 15561250048], [-7924213823951994829789477896052 : 8735625093986121050890513424 : 70673278488166002200994828211], ...
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-925	[-925, 0, 1, 0, 0] 49465822	0	-	-	-
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-930	[-930, 0, 1, 0, 0] 804357027	1	???	?	?
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-989	[-989, 0, 1, 0, 0] 60460106	0	-	-	-
-990	[-990, 0, 1, 0, 0] 970299027	0	-	-	-
-991	[-991, 0, 1, 0, 0] 973242298	1	[-1753950529759971963731374998179711082473423114297448263087561167507602723179915141245789356919952989630646083524405770208785652792918125654727136674990682736234100518753992084186761798442493602737588346821709896 : -9558510178649353136668018287419482726710513532460682786174779204551785588166722401694952745554648688869191671341112249300308656196776968202568422169959344359098868574525429681748632599208443727479628289729024 : 324707322877811281853952683168725125531651858946223809581622967670710623057027059408481325351889653703645709083829758512880803257867642177476523035555536359079064146270993590432450012501817485626712901681861849]	325.661108298528	[30683852670557210385672933495045991695244306309734084279086126352197915767751691742388396477766318851124344717130792848945150125842978480469697532896728515721477382591011992276205827272149218058088663918497902984 : -30956902474567817174175697556867449845175251866427562753458873985582124792805824765622940730310104158308756173026138572526892087173539498842395344225924343510274261269913090414469291668012569301404045052111424 : 5680474726728629724130433020213267265101429166971123250128199956275983441940375227799958163395506828726571784265289471110276759692031843668512748052157213330137138439281622369595331369718962158447479069648885121], ...
-992	[-992, 0, 1, 0, 0] 976191515	1	[252 : -64 : 1]	6.63094622704689	[-15876 : 16 : 63], ...
-993	[-993, 0, 1, 0, 0] 489573342	0	-	-	-
-994	[-994, 0, 1, 0, 0] 982107811	1	[-668652644775379554580391886018162873114830396288360519192917579405168845056773177626829319796931408539773880751174937932924935699478049989995474188662275163869460 : -14035819225956533144111503703119796988930538777611084560317605977011414669078097126143226761752794062767917612058739736601429538360761786442119344442768748580019721 : 32737580248258756223801431460471553036491374967233169759495466969059266556834288350963253722673160356245158630066195074223244213492078833744781740025091121856]	251.955959499150	[181116281929363175321405950895357629792767595512431464923044109491127248556977730265547488872760237551068607664823332731049923611189756490720639141111225612615211225 : -186140425668761579475205649569116524229193268443967699978826423553889168809005829267187097525768573160530034013595916384397792813469530166376655066277097663679116 : 8867547089297188919037181085304945188986849845239421430741819494895997644479469432659063925061552057862157087496774423565953439200595393315787908394704037602160], ...
-995	[-995, 0, 1, 0, 0] 26623646	1	[-146262837294 :, -770212616504625 : 5929741]	22.8129798777182	[712372744999440 : -1520847472725 : 2888078716], ...
-996	[-996, 0, 1, 0, 0] 12198123	1	[36410 : -1331 : 1000]	8.78001615779044	[-1205171 : 1210 : 33100], ...
-997	[-997, 0, 1, 0, 0] 9910270	0	-	-	-
-998	[-998, 0, 1, 0, 0] 994012019	0	-	-	-
-999	[-999, 0, 1, 0, 0] 997003026	1	[517337415436746536833555244265126506042395210396619364632178746366457307834306980828744300577535626955041930105894063569822766829073383114748252599475765952667348830577869824745279824814667884773611692079814484565169196295388696388310467771101016551267208437138728189076924 : -497030114168132112721782602982086423242843259024794831797473891351146392131484841661046426446758328121285860102418672944092950403714387949575259770795833900884159625699444435117808182984589048178255061603193839690464469809118900765730807229407131005309478049906908260094483 : 517316569639361637308486057105553009130640520956514045097705753160820685519753994127940775223753764613563136779474988862220530813748766685341698033348311331253472577149342390610690938439347839225797055060732427565976204416733787611755771329248095029387882431205455155029970423]	421.696893142399	[-5243033321743426372302347947328035016308259430632818230416790420877582578389649380662032601423693537897017925422340177793973087426784924960903002753192743827721489188633092564676577431479142346499402665083608743549032664684132609710061085745332659245839184305954772656432 : 5037022971366100005090287959811356069293018006108977837898938545142193277620882949816903293543647450940557346508748540956142089915314864842107980952912624115140299727790917573909263141976136908449550506570982686955744249558954459157233086893361561695301383908918482038967663 : 5242822056895676918501305572559301313347386283944912123559130039119100421871806910554007334656744827454832970753172710863714723543829050356606251595517842527169887083620547535268326367895936552090680899469664145110721581402778341895068755856567091016076830016493381649769964], ...
-1000	[-1000, 0, 1, 0, 0] 1000000027	0	-	-	-

楕円曲線E_n(n \in [-1000..-501])のMordel-Weil群のrankと生成元が求まらなかったのは、以下の15個である。
E_-590, E_-656, E_-703, E_-716, E_-728, E_-767, E_-780, E_-816, E_-818, E_-838
E_-902, E_-930, E_-938, E_-951, E_-968

[関連リンク]

[参考文献]

[1]Joseph H. Silverman, John Tate(著), 足立恒雄, 木田雅成, 小松啓一, 田谷久雄(訳), "楕円曲線論入門", シュプリンガー・フェアラーク東京, 1995, ISBN4-431-70683-6, {3900円}.
[2]Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.
[3]Dave Rusin, "For which values of n is a/b + b/c + c/a = n solvable?", 2003.08.01.
[4]Allan MacLeod, "x/y+y/z+z/x=N".
[5]AT&T Labs Research, "Integers expressible as (x^3+y^3+z^3)/xyz with positive integers x,y, and z.

Last Update: 2010.10.20

H.Nakao