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Rational Points on Elliptic Curves: u^3+v^3-3uv=n, y^2=x^3+9(4n+1)x^2+432n(n+1)x+1728(n+1)2 (n \in [2000..2100])

[2021.06.18]u^3+v^3-3uv=n, y^2=x^3+9(4n+1)x^2+432n(n+1)x+1728(n+1)2 (n \in [2000..2100])の有理点

■パラメータnを持つ楕円曲線
     Cn: u3+v3-3uv=n ----- (1)
を以下の標準形に変形する。
式の変形については、こちらを参照のこと。

     En: y2 = x3+9(4n+1)x2+432n(n+1)x+1728n(n+1)2 ------------- (2)

CnからEnへの有理変換φ:(u,v)-->(x,y)は、
     x=-12(n+1)(u+v)/(u+v+1),
     y=36(n+1)(u-v)/(u+v+1)
であり、EnからCnへの有理変換ψ:(x,y)-->(u,v)は、
     u=(-3x+y)/{6(x+12(n+1))},
     v=-(3x+y)/{6(x+12(n+1))}
である。

■n=2020..2030について、En, Cnの有理点を計算した結果をこちらにまとめた。今回、n=2000..2100についての結果に拡張した。

■n=2000..2100について、En, Cnの有理点を計算する。

楕円曲線Enのねじれ点群En(Q)torsは自明な群{O}である。
Cnは、無限遠点OC=[1,-1,0](有理点でもある)を持つ。

[pari/gpによる計算]
gp> for(n=2000,2100,e=ec(n);print("E_",n,"(Q)_{tors}=",elltors(e)))
E_2000(Q)_{tors}=[1, [], []]
E_2001(Q)_{tors}=[1, [], []]
E_2002(Q)_{tors}=[1, [], []]
E_2003(Q)_{tors}=[1, [], []]
E_2004(Q)_{tors}=[1, [], []]
E_2005(Q)_{tors}=[1, [], []]
E_2006(Q)_{tors}=[1, [], []]
E_2007(Q)_{tors}=[1, [], []]
E_2008(Q)_{tors}=[1, [], []]
E_2009(Q)_{tors}=[1, [], []]
E_2010(Q)_{tors}=[1, [], []]
E_2011(Q)_{tors}=[1, [], []]
E_2012(Q)_{tors}=[1, [], []]
E_2013(Q)_{tors}=[1, [], []]
E_2014(Q)_{tors}=[1, [], []]
E_2015(Q)_{tors}=[1, [], []]
E_2016(Q)_{tors}=[1, [], []]
E_2017(Q)_{tors}=[1, [], []]
E_2018(Q)_{tors}=[1, [], []]
E_2019(Q)_{tors}=[1, [], []]
E_2020(Q)_{tors}=[1, [], []]
E_2021(Q)_{tors}=[1, [], []]
E_2022(Q)_{tors}=[1, [], []]
E_2023(Q)_{tors}=[1, [], []]
E_2024(Q)_{tors}=[1, [], []]
E_2025(Q)_{tors}=[1, [], []]
E_2026(Q)_{tors}=[1, [], []]
E_2027(Q)_{tors}=[1, [], []]
E_2028(Q)_{tors}=[1, [], []]
E_2029(Q)_{tors}=[1, [], []]
E_2030(Q)_{tors}=[1, [], []]
E_2031(Q)_{tors}=[1, [], []]
E_2032(Q)_{tors}=[1, [], []]
E_2033(Q)_{tors}=[1, [], []]
E_2034(Q)_{tors}=[1, [], []]
E_2035(Q)_{tors}=[1, [], []]
E_2036(Q)_{tors}=[1, [], []]
E_2037(Q)_{tors}=[1, [], []]
E_2038(Q)_{tors}=[1, [], []]
E_2039(Q)_{tors}=[1, [], []]
E_2040(Q)_{tors}=[1, [], []]
E_2041(Q)_{tors}=[1, [], []]
E_2042(Q)_{tors}=[1, [], []]
E_2043(Q)_{tors}=[1, [], []]
E_2044(Q)_{tors}=[1, [], []]
E_2045(Q)_{tors}=[1, [], []]
E_2046(Q)_{tors}=[1, [], []]
E_2047(Q)_{tors}=[1, [], []]
E_2048(Q)_{tors}=[1, [], []]
E_2049(Q)_{tors}=[1, [], []]
E_2050(Q)_{tors}=[1, [], []]
E_2051(Q)_{tors}=[1, [], []]
E_2052(Q)_{tors}=[1, [], []]
E_2053(Q)_{tors}=[1, [], []]
E_2054(Q)_{tors}=[1, [], []]
E_2055(Q)_{tors}=[1, [], []]
E_2056(Q)_{tors}=[1, [], []]
E_2057(Q)_{tors}=[1, [], []]
E_2058(Q)_{tors}=[1, [], []]
E_2059(Q)_{tors}=[1, [], []]
E_2060(Q)_{tors}=[1, [], []]
E_2061(Q)_{tors}=[1, [], []]
E_2062(Q)_{tors}=[1, [], []]
E_2063(Q)_{tors}=[1, [], []]
E_2064(Q)_{tors}=[1, [], []]
E_2065(Q)_{tors}=[1, [], []]
E_2066(Q)_{tors}=[1, [], []]
E_2067(Q)_{tors}=[1, [], []]
E_2068(Q)_{tors}=[1, [], []]
E_2069(Q)_{tors}=[1, [], []]
E_2070(Q)_{tors}=[1, [], []]
E_2071(Q)_{tors}=[1, [], []]
E_2072(Q)_{tors}=[1, [], []]
E_2073(Q)_{tors}=[1, [], []]
E_2074(Q)_{tors}=[1, [], []]
E_2075(Q)_{tors}=[1, [], []]
E_2076(Q)_{tors}=[1, [], []]
E_2077(Q)_{tors}=[1, [], []]
E_2078(Q)_{tors}=[1, [], []]
E_2079(Q)_{tors}=[1, [], []]
E_2080(Q)_{tors}=[1, [], []]
E_2081(Q)_{tors}=[1, [], []]
E_2082(Q)_{tors}=[1, [], []]
E_2083(Q)_{tors}=[1, [], []]
E_2084(Q)_{tors}=[1, [], []]
E_2085(Q)_{tors}=[1, [], []]
E_2086(Q)_{tors}=[1, [], []]
E_2087(Q)_{tors}=[1, [], []]
E_2088(Q)_{tors}=[1, [], []]
E_2089(Q)_{tors}=[1, [], []]
E_2090(Q)_{tors}=[1, [], []]
E_2091(Q)_{tors}=[1, [], []]
E_2092(Q)_{tors}=[1, [], []]
E_2093(Q)_{tors}=[1, [], []]
E_2094(Q)_{tors}=[1, [], []]
E_2095(Q)_{tors}=[1, [], []]
E_2096(Q)_{tors}=[1, [], []]
E_2097(Q)_{tors}=[1, [], []]
E_2098(Q)_{tors}=[1, [], []]
E_2099(Q)_{tors}=[1, [], []]
E_2100(Q)_{tors}=[1, [], []]
time = 6 ms.

楕円曲線Enの有理点は、CremonaのmwrankまたはMAGMA Calculator(4-descent)を使って求める。
ただし、そのMordell-Weil群En(Q)がrank 1の場合は、pari/gpのellheegner()関数で求めても良い。

■n \in [2000..2100]について、結果をまとめると、以下のようになる。
・(OC以外の)有理点を持つもの(56個)
       C2000, C2001, C2002, C2005, C2006, C2007, C2008, C2010, C2013, C2016,
       C2020, C2022, C2023, C2024, C2025, C2027, C2028, C2029, C2030, C2033,
       C2034, C2035, C2036, C2037, C2038, C2040, C2043, C2047, C2049, C2050,
       C2051, C2053, C2055, C2056, C2058, C2060, C2061, C2063, C2065, C2069,
       C2070, C2072, C2073, C2075, C2078, C2080, C2085, C2088, C2090, C2091,
       C2092, C2093, C2094, C2095, C2097, C2099

・(OC以外の)有理点を持たない(整点も持たない)もの(45個)
       C2003, C2004, C2009, C2011, C2012, C2014, C2015, C2017, C2018, C2019,
       C2021, C2026, C2031, C2032, C2039, C2041, C2042, C2044, C2045, C2046,
       C2048, C2052, C2054, C2057, C2059, C2062, C2064, C2066, C2067, C2068,
       C2071, C2074, C2076, C2077, C2079, C2081, C2082, C2083, C2084, C2086,
       C2087, C2089, C2096, C2098, C2100

・(OC以外の)整点を持つもの(4個)
      C2001は、整点[11,10]を持つ。
      C2024は、整点[16,-14]を持つ。
      C2028は、整点[26,-26]を持つ。
      C2053は、整点[19,-18]を持つ。

n [a1,a2,a3,a4,a6],
conductor(En)		 rank
En(Q)		 En:y2z=x3+9(4n+1)x2z+432n(n+1)xz2+1728n(n+1)2z3
En(Q)/En(Q)torsの生成元
[x:y:z] 		En(Q)/En(Q)torsの生成元
の高さ		 Cn:u3+v3-3uvw=nw3
Cn(Q)/Cn(Q)torsの生成元
[u:v:w] 		n
2000 [0, 72009, 0, 1728864000, 13837827456000]
540270
 1 [-595635 : 2167120 : 27] 9.05746745440214 [3954025 : -380215 : 316134] 2000
2001 [0, 72045, 0, 1730592864, 13858587654912]
108162054
 1 [-22932 : 3276 : 1] 1.73537500733456 [11 : 10 : 1] 2001
2002 [0, 72081, 0, 1732322592, 13879368607104]
36090054
 1 [-101584540813012513705269108610171549526666585919996475392698095594947828 : 3608018145822046057392452074429017447202090131351448851041653248632803275 : 6005330173079471753336087604456696182036793604315055181064617150656] 109.832232715824 [434752418695675955389806600028836899531343321012382030802194170601960751 : -367029391487000946252960527622055866513565597065717713873728773538662199 : 28506383484750446238611395367033066603179856768880127292914028158813192] 2002
2003 [0, 72117, 0, 1734053184, 13900170322944]
54189162
 0 - - - 2003
2004 [0, 72153, 0, 1735784640, 13920992812800]
54243270
 0 - - - 2004
2005 [0, 72189, 0, 1737516960, 13941836087040]
108594810
 1 [480530766479088 : 688150099602254323 : 242970624] 27.3822154124350 [686708507302817059 : -689591691901691587 : 2918277332040096] 2005
2006 [0, 72225, 0, 1739250144, 13962700156032]
12078126
 1 [8650018425 : 3971171698858 : 421875] 19.3561859861355 [3945221643583 : -3997121754133 : 112862735550] 2006
2007 [0, 72261, 0, 1740984192, 13983585030144]
3022542
 1 [-23098638558190651838569589605 : 553238106013896476296529407562 : 1260825248336972702985125] 45.7815309758279 [622534021688468431812238176377 : -483942190339324520780820638747 : 43693239754422254475359894370] 2007
2008 [0, 72297, 0, 1742719104, 14004490719744]
3889998
 1 [-23016 : 504 : 1] 2.66008857137761 [138 : 136 : 13] 2008
2009 [0, 72333, 0, 1744454880, 14025417235200]
15575490
 0 - - - 2009
2010 [0, 72369, 0, 1746191520, 14046364586880]
109136970
 1 [-5876419036 : 126193900139 : 314432] 17.1730466858905 [143823157247 : -108564643031 : 10268723928] 2010
2011 [0, 72405, 0, 1747929024, 14067332785152]
18207594
 0 - - - 2011
2012 [0, 72441, 0, 1749667392, 14088321840384]
54677106
 0 - - - 2012
2013 [0, 72477, 0, 1751406624, 14109331762944]
109462914
 1 [-2879605 : 1544158 : 125] 10.8323267020059 [10182973 : 7094657 : 848370] 2013
2014 [0, 72513, 0, 1753146720, 14130362563200]
109571670
 0 - - - 2014
2015 [0, 72549, 0, 1754887680, 14151414251520]
761670
 0 - - - 2015
2016 [0, 72585, 0, 1756629504, 14172486838272]
762426
 1 [-2708680 : 15539912 : 125] 9.39591731058065 [2958244 : -926734 : 237615] 2016
2017 [0, 72621, 0, 1758372192, 14193580333824]
109898262
 0 - - - 2017
2018 [0, 72657, 0, 1760115744, 14214694748544]
110007234
 0 - - - 2018
2019 [0, 72693, 0, 1761860160, 14235830092800]
55058130
 0 - - - 2019
2020 [0, 72729, 0, 1763605440, 14256986376960]
18370890
 1 [-282621828033 : 13580911254862 : 17779581] 20.2460145951678 [14428776738961 : -12733045770763 : 891411422274] 2020
2021 [0, 72765, 0, 1765351584, 14278163611392]
110334474
 0 - - - 2021
2022 [0, 72801, 0, 1767098592, 14299361806464]
6496686
 1 [-30270813685369445720 : 28773327956015467441 : 1312213091392000] 30.8556946380325 [119585769012123804601 : 62039113100092869719 : 9506827927576477680] 2022
2023 [0, 72837, 0, 1768846464, 14320580972544]
1625778
 1 [-272764348764 : 338101366033 : 11852352] 17.1198498745814 [1156394412325 : 480191680259 : 90633459672] 2023
2024 [0, 72873, 0, 1770595200, 14341821120000]
22770
 1 [-16200 : 729000 : 1] 0.70634945873924 [16 : -14 : 1] 2024
2025 [0, 72909, 0, 1772344800, 14363082259200]
30390
 1 [-18909 : 396900 : 1] 6.53107149996338 [50403 : -37797 : 3602] 2025
2026 [0, 72945, 0, 1774095264, 14384364400512]
110880954
 0 - - - 2026
2027 [0, 72981, 0, 1775846592, 14405667554304]
4268862
 1 [-2903680 : 683488 : 125] 6.26004268223425 [22583 : 19297 : 1995] 2027
2028 [0, 73017, 0, 1777598784, 14426991730944]
4273074
 2 [-22997 : 35074 : 1],
[0 : 3798288 : 1]
 6.42820715523165,
0.57273790232839
 [104065 : 33917 : 8106],
[26 : -26 : 1]
 2028
2029 [0, 73053, 0, 1779351840, 14448336940800]
37069830
 1 [-11172 : 1514268 : 1] 5.21028671071838 [3071 : -2938 : 157] 2029
2030 [0, 73089, 0, 1781105760, 14469703194240]
111319110
 1 [-953233610918881541254694039221515642635411990563975974344670008492439526702143413562514766216890397990217053251165736802313212789373226540536480420772458449683000690680037581736999860318089918964479 : 991061691906829633530723549108825288740858261941908355976772335649768499710874816216521854554267813012741688353593680837173330468281714731006659141445614043784609485252141032234043841246493118639474 : 41191087645563362723749302508057851797086585256188999049308316010677534963994625616042573512203090006422983172177548957875790513830902442958617051170728814655354501120362260016768374949157409803] 304.869542068769 [3850762524663474257294805666773372216647094233633836279010782361127087079817305056904066153204939006983392848107090891244112968836401394352616100403762989392833611557292253777445043422200762875532911 : 1868639140849814990233358568555721639165377709750019567057237689827550080395555424471022444096403380957909471399903529569766307899837964890602782120871761305264392586787971712976955739707776638253963 : 304053463072732410291143769029221928179093591799173862910433615918760132642001611710049012535139869877943355726872918394213321682281166795505606102163265326583795263752588516350073843656646836525422] 2030
2031 [0, 73125, 0, 1782860544, 14491090501632]
13928598
 0 - - - 2031
2032 [0, 73161, 0, 1784616192, 14512498873344]
13942314
 0 - - - 2032
2033 [0, 73197, 0, 1786372704, 14533928319744]
12405366
 1 [-1114284510 : 2522264190209 : 729000] 19.0691293405582 [2525607043739 : -2518921336679 : 100074884940] 2033
2034 [0, 73233, 0, 1788130080, 14555378851200]
12417570
 1 [-393830947765337672485172172029290385684296898 : 2491721327473531799171188502166206909195772775 : 18132188318799222898786981278651200738776] 69.9650919223222 [3673214170769544816626705018254078066248663469 : -1310228484177518781715671986078335752142882081 : 293742545878436104219235464772231618139678132] 2034
2035 [0, 73269, 0, 1789888320, 14576850478080]
55934010
 1 [-629088 : 301760 : 27] 7.38879592952466 [68407 : 49547 : 5733] 2035
2036 [0, 73305, 0, 1791647424, 14598343210752]
55988982
 1 [-1009510040 : 51669783557 : 64000] 12.1818846757116 [80557163 : -71636603 : 4903440] 2036
2037 [0, 73341, 0, 1793407392, 14619857059584]
112087962
 1 [-42846844310335 : 117234541373912 : 1883652875] 22.0893593600773 [245775074304917 : 11305991557093 : 19318622403990] 2037
2038 [0, 73377, 0, 1795168224, 14641392034944]
37399338
 1 [-174766380 : 1049290929 : 8000] 12.4883038508963 [174843341 : -58332421 : 13985080] 2038
2039 [0, 73413, 0, 1796929920, 14662948147200]
28077030
 0 - - - 2039
2040 [0, 73449, 0, 1798692480, 14684525406720]
28104570
 1 [-23048 : 42328 : 1] 5.90190330236311 [13934 : 3352 : 1083] 2040
2041 [0, 73485, 0, 1800455904, 14706123823872]
112528494
 0 - - - 2041
2042 [0, 73521, 0, 1802220192, 14727743409024]
12515418
 0 - - - 2042
2043 [0, 73557, 0, 1803985344, 14749384172544]
6263838
 1 [-23360 : 16352 : 1] 2.34833988157083 [37 : 23 : 3] 2043
2044 [0, 73593, 0, 1805751360, 14771046124800]
56429730
 0 - - - 2044
2045 [0, 73629, 0, 1807518240, 14792729276160]
112969890
 0 - - - 2045
2046 [0, 73665, 0, 1809285984, 14814433636992
113080374
 0 - - - 2046
2047 [0, 73701, 0, 1811054592, 14836159217664]
36846
 1 [-2929245 : 1518426 : 125] 9.19475186623962 [1145129 : 807701 : 95170] 2047
2048 [0, 73737, 0, 1812824064, 14857906028544]
110646
 0 - - - 2048
2049 [0, 73773, 0, 1814594400, 14879674080000]
22682430
 1 [-76280115 : 281229652 : 3375] 13.3705939951648 [510069997 : -52389307 : 40469310] 2049
2050 [0, 73809, 0, 1816365600, 14901463382400]
22704570
 1 [-498020965811220088950 : 237367521073899885655 : 21215744852461704] 31.8800495257370 [1731430418507560152505 : 1256695376359760381195 : 144845678985404219388] 2050
2051 [0, 73845, 0, 1818137664, 14923273946112]
701442
 1 [-20672 : 247456 : 1] 4.07675165201353 [509 : -305 : 39] 2051
2052 [0, 73881, 0, 1819910592, 14945105781504]
78014
 0 - - - 2052
2053 [0, 73917, 0, 1821684384, 14966958898944]
113855274
 1 [-12324 : 1367964 : 1] 2.00429414262201 [19 : -18 : 1] 2053
2054 [0, 73953, 0, 1823459040, 14988833308800]
113966190
 0 - - - 2054
2055 [0, 73989, 0, 1825234560, 15010729021440]
28519290
 1 [-516668516 : 228470201 : 21952] 14.2866674865953 [1778475749 : 1321535347 : 149587368] 2055
2056 [0, 74025, 0, 1827010944, 15032646047232]
865062
 1 [2055720 : 1042444728 : 125] 10.3646892147881 [14392744 : -14564054 : 428435] 2056
2057 [0, 74061, 0, 1828788192, 15054584396544]
212058
 0 - - - 2057
2058 [0, 74097, 0, 1830566304, 15076544079744]
2334906
 2 [-22337 : -111356 : 1],
[-635019 : -497168 : 27]
 6.77626591308938,
8.54608264859067
 [-44345 : 178367 : 14226],
[1407889 : 2402225 : 192582]
 2058
2059 [0, 74133, 0, 1832345280, 15098525107200]
57260790
 0 - - - 2059
2060 [0, 74169, 0, 1834125120, 15120527489280]
6368490
 1 [-64742202 : 28776033 : 2744] 10.7049205985529 [8259357 : 6127799 : 693868] 2060
2061 [0, 74205, 0, 1835905824, 15142551236352]
12749346
 1 [-753918282403442555890355765691365673716740 : 14499323616860137023267814200092994976123687 : 38594393351178157201058607221986904000] 66.3958839201597 [16761078464070464690938881497167091997273907 : -12237568769649809355596746903018897954973467 : 1206368320068658595355830468456869673155560] 2061
2062 [0, 74241, 0, 1837687392, 15164596358784]
114855462
 0 - - - 2062
2063 [0, 74277, 0, 1839469824, 15186662866944]
14370858
 1 [-187222 : 284273 : 8] 6.60040220705451 [19673 : 6451 : 1524] 2063
2064 [0, 74313, 0, 1841253120, 15208750771200]
14384790
 0 - - - 2064
2065 [0, 74349, 0, 1843037280, 15230860081920
38396610
 1 [31164693 : 13488865200 : 343] 13.9085171030448 [1488374569 : -1509151031 : 26445566] 2065
2066 [0, 74385, 0, 1844822304, 15252990809472]
115301394
 0 - - - 2066
2067 [0, 74421, 0, 1846608192, 15275142964224]
57706506
 0 - - - 2067
2068 [0, 74457, 0, 1848394944, 15297316556544]
57762342
 0 - - - 2068
2069 [0, 74493, 0, 1850182560, 15319511596800]
12848490
 1 [6737906844 : 6932967837228 : 6859] 16.1287508899294 [32003491281 : -32190655360 : 191896789] 2069
2070 [0, 74529, 0, 1851971040, 15341728095360]
12860910
 1 [-11212576986 : 10498276729 : 474552] 16.3767901510745 [44136007687 : 23139454229 : 3485935908] 2070
2071 [0, 74565, 0, 1853760384, 15363966062592]
28965006
 0 - - - 2071
2072 [0, 74601, 0, 1855550592, 15386225508864]
28992978
 1 [-68405160 : 1053658504 : 3375] 12.1621467922738 [157359248 : -106055378 : 11663505] 2072
2073 [0, 74637, 0, 1857341664, 15408506444544]
116083854
 1 [-23180 : 61732 : 1] 3.72872171315531 [269 : 16 : 21] 2073
2074 [0, 74673, 0, 1859133600, 15430808880000]
7746390
 0 - - - 2074
2075 [0, 74709, 0, 1860926400, 15453132825600]
11630790
 1 [-217825440 : 346860800 : 9261] 10.5921958386690 [31260535 : 9581735 : 2415861] 2075
2076 [0, 74745, 0, 1862720064, 15475478291712]
58210002
 0 - - - 2076
2077 [0, 74781, 0, 1864514592, 15497845288704]
116532162
 0 - - - 2077
2078 [0, 74817, 0, 1866309984, 15520233826944]
1440054
 1 [-181062 : 855603 : 8] 4.85175217506516 [2467 : -551 : 196] 2078
2079 [0, 74853, 0, 1868106240, 15542643916800]
10010
 0 - - - 2079
2080 [0, 74889, 0, 1869903360, 15565075568640]
7304310
 1 [-539376 : 9520496 : 27] 8.64243857168104 [1392328 : -987796 : 101151] 2080
2081 [0, 74925, 0, 1871701344, 15587528792832]
116981334
 0 - - - 2081
2082 [0, 74961, 0, 1873500192, 15610003599744]
117093762
 0 - - - 2082
2083 [0, 74997, 0, 1875299904, 15632499999744]
19534374
 0 - - - 2083
2084 [0, 75033, 0, 1877100480, 15655018003200]
58659390
 0 - - - 2084
2085 [0, 75069, 0, 1878901920, 15677557620480]
117431370
 1 [8542763200159064308126072 : 3797874156290170635814686067 : 83085958298639166976] 42.3419399698703 [3772245866689693442890307851 : -3823502445890647828739064283 : 63735425449743599615215824] 2085
2086 [0, 75105, 0, 1880704224, 15700118861952]
117544014
 0 - - - 2086
2087 [0, 75141, 0, 1882507392, 15722701737984]
3268242
 0 - - - 2087
2088 [0, 75177, 0, 1884311424, 15745306258944]
3271374
 1 [-1827880 : 133428232 : 125] 10.6948597539276 [17363984 : -15993074 : 979215] 2088
2089 [0, 75213, 0, 1886116320, 15767932435200]
117882270
 0 - - - 2089
2090 [0, 75249, 0, 1887922080, 15790580277120]
117995130
 1 [-9955677880413656773839429890268695246890328305579825996973487274042685374899161168961445366308081152054143751655763392253863416892097912917386809617319617108254687885328125533649775683701317 : 5328253172868219425950073219260986386248865922664526895178483803753741957557438615034413479872873016640847846220747373098980985223299034851679573115189740807112964576093358693928391519962350 : 416015763441052613314798883832900557264442120647358310472869435403646765444437008577773737375007233328326521575489262624349700147544039819468620673705520398821527317153141515857117855033] 292.574822513327 [35195286814109189747468362890067072126919850839404004886098945625881798082254922121918749578797116472803279101188037549860571235899592773603840001967148592131877028232077735294877718571066301 : 24538780468372750895568216451545099354422118994074951095741978018374314167140044891849922619051370439521583408746542803662609265452994703900480855736769110517651099079891017907020935531141601 : 2897937935095412396733022217198673215934320314222132376470515594633715581795913501632319511433602079721351966298479113097915555260462805396318921963795804433850453340071008293422153208720314] 2090
2091 [0, 75285, 0, 1889728704, 15813249795072]
59054022
 2 [-23012 : -89956 : 1],
[-3313728 : 2148383584 : 729]
 2.43038691436577,
7.76217005209982
 [-5 : 38 : 3],
[128963 : -127775 : 5373]
 2091
2092 [0, 75321, 0, 1891536192, 15835940999424]
19703502
 1 [-17472 : 668304 : 1] 2.65019104832619 [110 : -94 : 7] 2092
2093 [0, 75357, 0, 1893344544, 15858653900544]
118334034
 1 [-573999 : 6466328 : 27] 10.6481054060250 [8188325 : -4744331 : 626742] 2093
2094 [0, 75393, 0, 1895153760, 15881388508800]
118447110
 2 [-7778939 : 40521790 : 343],
[2680228 : 1110211325 : 64]
 11.9952862612534,
13.8962639736259
 [63858607 : -17184973 : 5064486],
[1102170641 : -1118252009 : 25735128]
 2094
2095 [0, 75429, 0, 1896963840, 15904144834560]
14820030
 1 [1639977401432242276194619038665595 : 691485666525928101224826820961682904 : 47034455835841165448811518625] 55.0284446825324 [686565734321631374396242963845686119 : -696405598730224828053410678077679689 : 16937928207691915617378758130729570] 2095
2096 [0, 75465, 0, 1898774784, 15926922888192]
1648242
 0 - - - 2096
2097 [0, 75501, 0, 1900586592, 15949722680064]
13198518
 1 [-12453088837969035445065495780 : 29604688351616703231909338743 : 528980103028305539352000] 44.2481557057324 [66963954865523809567105826083 : 7754578162290403103287148597 : 5187085415229508881962737320] 2097
2098 [0, 75537, 0, 1902399264, 15972544220544]
118899954
 0 - - - 2098
2099 [0, 75573, 0, 1904212800, 15995387520000]
11901330
 1 [-51833600 : 118412000 : 2197] 6.81384347946915 [48913 : 6623 : 3783] 2099
2100 [0, 75609, 0, 1906027200, 16018252588800]
11912670
 0 - - - 2100


[参考文献]

Last Update: 2024.01.09
H.Nakao

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