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Rational Points on Ellptic Curves: y^2=x^3+n (n \in [7001..8000])


[2021.11.29]y^2=x^3+n (n \in [7001..8000])の有理点



■n=7001,7002,...,8000に対して、
     En: y2 = x3+n
の有理点を求める。

■n=7001,7002,...,8000に対して、楕円曲線Enのねじれ点群En(Q)torsを計算すると、
     En(Q)tors=Z/2Z if n=8000
                 =Z/3Z if n=7056,7225,7396,7569,7744,7921
                 ={O} if nが平方数,立方数でない
となる。

[pari/gpによる計算]
gp> ec(n)=ellinit([0,0,0,0,n]);
gp> tg(n1,n2)=local(e,g,o);for(i=n1,n2,e=ec(i);g=elltors(e);o=g[1];if(o>1,print("E_",i,"(Q)_{tors}=",g)));
gp> tg(7001,8000)
E_7056(Q)_{tors}=[3, [3], [[0, 84]]]
E_7225(Q)_{tors}=[3, [3], [[0, 85]]]
E_7396(Q)_{tors}=[3, [3], [[0, 86]]]
E_7569(Q)_{tors}=[3, [3], [[0, 87]]]
E_7744(Q)_{tors}=[3, [3], [[0, 88]]]
E_7921(Q)_{tors}=[3, [3], [[0, 89]]]
E_8000(Q)_{tors}=[2, [2], [[-20, 0]]]
time = 7 ms.

■楕円曲線EnのMordell-Weil群のrankと生成元を求めると以下のようになる。

- E~n: Y2Z=X3+nZ3
En: y2=x3+n
-
n [a1,a2,a3,a4,a6]
j(En)
Conductor of En
En(Q)tors
En(Q)torsの生成元
rank(En(Q)) En(Q)/En(Q)tors
の生成元 [X:Y:Z]
En(Q)/En(Q)tors
の生成元の高さ
En(Z)
Enの整点[x,y]
(where y >= 0)
n
7001 [0, 0, 0, 0, 7001]
0
1764504036
{O}
-
0 - - - 7001
7002 [0, 0, 0, 0, 7002]
0
2353344192
{O}
-
2 [63 : 507 : 1],
[343 : 6353 : 1]
3.60044704783801,
5.87205601721438
[63, 507], [343, 6353] 7002
7003 [0, 0, 0, 0, 7003]
0
7062049296
{O}
-
1 [-19 : 12 : 1] 2.95047957345636
[-19, 12] 7003
7004 [0, 0, 0, 0, 7004]
0
1324512432
{O}
-
1 [70070 : -13923982 : 166375] 10.7351049294791
- 7004
7005 [0, 0, 0, 0, 7005]
0
5299562700
{O}
-
0 - - - 7005
7006 [0, 0, 0, 0, 7006]
0
21204303552
{O}
-
1 [-48264098059785 : -2922124184519813 : 34917687019125] 24.2037195596265
- 7006
7007 [0, 0, 0, 0, 7007]
0
432864432
{O}
-
0 - - - 7007
7008 [0, 0, 0, 0, 7008]
0
82876608
{O}
-
2 [97 : 959 : 1],
[-142 : 301 : 8]
4.69693317609235,
4.87797795536516
[97, 959] 7008
7009 [0, 0, 0, 0, 7009]
0
5305616748
{O}
-
1 [6 : 85 : 1] 2.96918963288114
[6, 85] 7009
7010 [0, 0, 0, 0, 7010]
0
7076174400
{O}
-
1 [191 : 2641 : 1] 5.31402435400320
[191, 2641] 7010
7011 [0, 0, 0, 0, 7011]
0
2359397808
{O}
-
1 [173301695 : 1443020444 : 15813251] 14.5277019466580
- 7011
7012 [0, 0, 0, 0, 7012]
0
110628324
{O}
-
2 [-16 : 54 : 1],
[173 : 2277 : 1]
2.00696101438319,
4.67219685329255
[-16, 54], [[173, 2277] 7012
7013 [0, 0, 0, 0, 7013]
0
5311674252
{O}
-
0 - - - 7013
7014 [0, 0, 0, 0, 7014]
0
21252756672
{O}
-
1 [-5 : 83 : 1] 3.42302900246674
[-5, 83] 7014
7015 [0, 0, 0, 0, 7015]
0
21258817200
{O}
-
1 [9 : 88 : 1] 3.45554017246667
[9, 88] 7015
7016 [0, 0, 0, 0, 7016]
0
1329054912
{O}
-
0 - - - 7016
7017 [0, 0, 0, 0, 7017]
0
5317735212
{O}
-
1 [-46 : 661 : 8] 4.81210819078813
- 7017
7018 [0, 0, 0, 0, 7018]
0
175843008
{O}
-
1 [-483 : 971 : 27] 5.69024423715581
- 7018
7019 [0, 0, 0, 0, 7019]
0
7094355984
{O}
-
3 [13 : 96 : 1],
[425 : 8762 : 1],
[5074 : 127787 : 8]
2.95459930088452,
6.07980815467824,
7.85730724316323
[13, 96], [425, 8762] 7019
7020 [0, 0, 0, 0, 7020]
0
608400
{O}
-
0 - - - 7020
7021 [0, 0, 0, 0, 7021]
0
1774599876
{O}
-
0 - - - 7021
7022 [0, 0, 0, 0, 7022]
0
21301265088
{O}
-
1 [-28 : 5363 : 64] 6.18703389074021
- 7022
7023 [0, 0, 0, 0, 7023]
0
21307332528
{O}
-
1 [-69 : 2260 : 27] 5.61417948684802
- 7023
7024 [0, 0, 0, 0, 7024]
0
83255472
{O}
-
1 [-636 : 4975 : 64] 6.21774388580776
- 7024
7025 [0, 0, 0, 0, 7025]
0
213194700
{O}
-
0 - - - 7025
7026 [0, 0, 0, 0, 7026]
0
21325540032
{O}
-
0 - - - 7026
7027 [0, 0, 0, 0, 7027]
0
21331610928
{O}
-
0 - - - 7027
7028 [0, 0, 0, 0, 7028]
0
111133764
{O}
-
2 [-19 : 13 : 1],
[2258 : 37941 : 8]
3.50082250321534,
6.52155930398053
[-19, 13] 7028
7029 [0, 0, 0, 0, 7029]
0
592882092
{O}
-
3 [3 : 84 : 1],
[15 : 102 : 1],
[-17 : 46 : 1]
2.68637706405679,
2.80195422613234,
3.48764992695806
[-17, 46], [3, 84], [15, 102], [6675, 545352] 7029
7030 [0, 0, 0, 0, 7030]
0
7116609600
{O}
-
1 [3655 : 22371 : 125] 6.46773426387426
- 7030
7031 [0, 0, 0, 0, 7031]
0
21355903152
{O}
-
1 [43948995 : 800037116 : 132651] 13.7023017689844
- 7031
7032 [0, 0, 0, 0, 7032]
0
1335123648
{O}
-
1 [683353639374 : -7775557580440 : 5295319407] 19.1854664259430
- 7032
7033 [0, 0, 0, 0, 7033]
0
5342013612
{O}
-
0 - - - 7033
7034 [0, 0, 0, 0, 7034]
0
21374131392
{O}
-
0 - - - 7034
7035 [0, 0, 0, 0, 7035]
0
21380209200
{O}
-
2 [2 : 671 : 8],
[-1276699 : -1795396 : 68921]
4.80130316352717,
10.9250828393974
- 7035
7036 [0, 0, 0, 0, 7036]
0
1336642992
{O}
-
0 - - - 7036
7037 [0, 0, 0, 0, 7037]
0
1782697284
{O}
-
0 - - - 7037
7038 [0, 0, 0, 0, 7038]
0
2377605312
{O}
-
1 [12009 : -253277 : 27] 8.32130870054353
- 7038
7039 [0, 0, 0, 0, 7039]
0
7134843024
{O}
-
1 [644 : 5745 : 64] 5.69090283059516
- 7039
7040 [0, 0, 0, 0, 7040]
0
5227200
{O}
-
1 [2596 : 111736 : 1331] 6.82639516275280
- 7040
7041 [0, 0, 0, 0, 7041]
0
5354173548
{O}
-
2 [22 : 133 : 1],
[-188784 : -1544669 : 19683]
3.19725229201100,
9.57369690868940
[22, 133] 7041
7042 [0, 0, 0, 0, 7042]
0
21422778048
{O}
-
1 [183 : 2477 : 1] 5.27404615828850
[183, 2477] 7042
7043 [0, 0, 0, 0, 7043]
0
21428862768
{O}
-
0 - - - 7043
7044 [0, 0, 0, 0, 7044]
0
334921068
{O}
-
1 [-9468 : 50786 : 729] 6.93343583272025
- 7044
7045 [0, 0, 0, 0, 7045]
0
5360258700
{O}
-
1 [243594 : 1731677 : 17576] 10.0335825467851
- 7045
7046 [0, 0, 0, 0, 7046]
0
7149040704
{O}
-
1 [4046 : 230385 : 2744] 8.14525909080398
- 7046
7047 [0, 0, 0, 0, 7047]
0
3269808
{O}
-
1 [14989 : 188540 : 2197] 8.56840263446172
- 7047
7048 [0, 0, 0, 0, 7048]
0
447068736
{O}
-
1 [2 : 84 : 1] 2.17491165685660
[2, 84] 7048
7049 [0, 0, 0, 0, 7049]
0
5366347308
{O}
-
0 - - - 7049
7050 [0, 0, 0, 0, 7050]
0
858859200
{O}
-
2 [1202 : 14749 : 8],
[2795 : 16877 : 125]
6.47733677220495,
6.88533063265162
- 7050
7051 [0, 0, 0, 0, 7051]
0
21477571632
{O}
-
1 [-83264357008406940 : 2592572463245846269 : 30917752140456000] 28.7319480531084
- 7051
7052 [0, 0, 0, 0, 7052]
0
1342729008
{O}
-
0 - - - 7052
7053 [0, 0, 0, 0, 7053]
0
5372439372
{O}
-
1 [7 : 86 : 1] 3.44006169343856
[7, 86] 7053
7054 [0, 0, 0, 0, 7054]
0
21495851712
{O}
-
0 - - - 7054
7055 [0, 0, 0, 0, 7055]
0
7167315600
{O}
-
3 [1 : 84 : 1],
[-19 : 14 : 1],
[4605722 : 27736457 : 195112]
2.86705238266209,
3.50193833824714,
11.8113276147091
[-19, 14], [1, 84] 7055
7056 [0, 0, 0, 0, 7056]
0
11907
Z/3Z
[0, 84]
0 - - - 7056
7057 [0, 0, 0, 0, 7057]
0
1792844964
{O}
-
4 [-1 : 84 : 1],
[-12 : 73 : 1],
[-18 : 35 : 1],
[48 : 343 : 1]
2.86712319497953,
2.99588131812732,
3.03328345806021,
3.66100616792524
[-18, 35], [-12, 73], [-1, 84], [8, 87], [14, 99], [48, 343], [68, 567], [107, 1110], [354, 6661], [939, 28774], [1674, 68491] 7057
7058 [0, 0, 0, 0, 7058]
0
21520237248
{O}
-
1 [4781463770934081 : -37366769361375415 : 80974801992447] 25.6318101586375
- 7058
7059 [0, 0, 0, 0, 7059]
0
21526335792
{O}
-
1 [-7919306165 : -39235845674 : 582182875] 16.9223878929547
- 7059
7060 [0, 0, 0, 0, 7060]
0
336444300
{O}
-
0 - - - 7060
7061 [0, 0, 0, 0, 7061]
0
5384633868
{O}
-
1 [-37265 : 2048884 : 24389] 10.1517370721299
- 7061
7062 [0, 0, 0, 0, 7062]
0
21544636608
{O}
-
2 [15580038435 : -194126629823 : 100544625],
[514785028498 : 7223810360963 : 2616662152]
17.4035986560671,
19.7987197380803
[12259, 1357321] 7062
7063 [0, 0, 0, 0, 7063]
0
21550738608
{O}
-
0 - - - 7063
7064 [0, 0, 0, 0, 7064]
0
449100864
{O}
-
1 [-2 : 84 : 1] 2.17547697093386
[-2, 84] 7064
7065 [0, 0, 0, 0, 7065]
0
598970700
{O}
-
2 [19 : 118 : 1],
[-142 : 307 : 8]
3.60411151635613,
4.88036258042512
[19, 118] 7065
7066 [0, 0, 0, 0, 7066]
0
7189683264
{O}
-
1 [47 : 333 : 1] 3.55842962947396
[47, 333] 7066
7067 [0, 0, 0, 0, 7067]
0
21575155248
{O}
-
1 [-7 : 82 : 1] 3.43309262973622
[-7, 82] 7067
7068 [0, 0, 0, 0, 7068]
0
1348828848
{O}
-
1 [157 : 1969 : 1] 5.13164714524098
[157, 1969] 7068
7069 [0, 0, 0, 0, 7069]
0
5396842188
{O}
-
0 - - - 7069
7070 [0, 0, 0, 0, 7070]
0
21593476800
{O}
-
0 - - - 7070
7071 [0, 0, 0, 0, 7071]
0
21599585712
{O}
-
0 - - - 7071
7072 [0, 0, 0, 0, 7072]
0
84397248
{O}
-
0 - - - 7072
7073 [0, 0, 0, 0, 7073]
0
1800983844
{O}
-
2 [-8 : 81 : 1],
[26 : 157 : 1]
2.42638955534220,
3.27489673437063
[-8, 81], [26, 157], [31, 192] 7073
7074 [0, 0, 0, 0, 7074]
0
9884736
{O}
-
2 [210 : 1269 : 8],
[-805 : 10313 : 125]
3.48020746877147,
6.64998311599399
- 7074
7075 [0, 0, 0, 0, 7075]
0
288320400
{O}
-
0 - - - 7075
7076 [0, 0, 0, 0, 7076]
0
337970988
{O}
-
0 - - - 7076
7077 [0, 0, 0, 0, 7077]
0
5409064332
{O}
-
2 [2 : 673 : 8],
[-2549512 : -11130923 : 175616]
4.80328719161261,
11.5246016130795
- 7077
7078 [0, 0, 0, 0, 7078]
0
21642372288
{O}
-
1 [6702606105 : 201518865937 : 7414875] 17.3658272568757
- 7078
7079 [0, 0, 0, 0, 7079]
0
21648488112
{O}
-
0 - - - 7079
7080 [0, 0, 0, 0, 7080]
0
1353412800
{O}
-
0 - - - 7080
7081 [0, 0, 0, 0, 7081]
0
5415180588
{O}
-
1 [5250 : 73243 : 27] 7.06613068997467
- 7081
7082 [0, 0, 0, 0, 7082]
0
7222280256
{O}
-
1 [-2850664388 : -11147466433 : 187149248] 16.1768470575565
- 7082
7083 [0, 0, 0, 0, 7083]
0
2408106672
{O}
-
2 [-3 : 84 : 1],
[-1570632 : -14013705 : 175616]
2.68827834244818,
10.7610449011788
[-3, 84] 7083
7084 [0, 0, 0, 0, 7084]
0
451647504
{O}
-
3 [-10 : 78 : 1],
[-19 : 15 : 1],
[-249 : 2143 : 27]
2.20571667715706,
2.95382621318543,
5.64140455208044
[-19, 15], [-10, 78], [78, 694], [40958, 8289114] 7084
7085 [0, 0, 0, 0, 7085]
0
5421300300
{O}
-
0 - - - 7085
7086 [0, 0, 0, 0, 7086]
0
21691323072
{O}
-
1 [-2573750 : -13504597 : 195112] 11.5871078275948
- 7086
7087 [0, 0, 0, 0, 7087]
0
21697445808
{O}
-
1 [119781 : -1536614 : 729] 9.56829017696718
- 7087
7088 [0, 0, 0, 0, 7088]
0
84779568
{O}
-
0 - - - 7088
7089 [0, 0, 0, 0, 7089]
0
5427423468
{O}
-
1 [208 : 3001 : 1] 4.93233933279316
[208, 3001] 7089
7090 [0, 0, 0, 0, 7090]
0
21715819200
{O}
-
0 - - - 7090
7091 [0, 0, 0, 0, 7091]
0
7240648464
{O}
-
1 [654670453 : 7084521900 : 81182737] 15.0424107040777
- 7091
7092 [0, 0, 0, 0, 7092]
0
37722348
{O}
-
2 [194 : 1169 : 8],
[853 : 24913 : 1]
5.09006369647607,
6.76261359727719
[853, 24913] 7092
7093 [0, 0, 0, 0, 7093]
0
1811183364
{O}
-
0 - - - 7093
7094 [0, 0, 0, 0, 7094]
0
21740329152
{O}
-
1 [-303 : 2035 : 27] 5.65243941532170
- 7094
7095 [0, 0, 0, 0, 7095]
0
21746458800
{O}
-
1 [1814551128842106216423032226838460951 : 17299196459545012723948791973395444046 : 20158591512508797463464319841191931] 57.2916558658114
- 7095
7096 [0, 0, 0, 0, 7096]
0
1359536832
{O}
-
1 [-15 : 61 : 1] 3.47771043698850
[-15, 61] 7096
7097 [0, 0, 0, 0, 7097]
0
5439680172
{O}
-
1 [-13 : 70 : 1] 3.46556215289422
[-13, 70] 7097
7098 [0, 0, 0, 0, 7098]
0
128786112
{O}
-
0 - - - 7098
7099 [0, 0, 0, 0, 7099]
0
21770986032
{O}
-
0 - - - 7099
7100 [0, 0, 0, 0, 7100]
0
18147600
{O}
-
2 [-14 : 66 : 1],
[5 : 85 : 1]
2.22925227969851,
2.35690782223389
[-14, 66], [5, 85], [10, 90], [1210, 42090] 7100
7101 [0, 0, 0, 0, 7101]
0
7470252
{O}
-
0 - - - 7101
7102 [0, 0, 0, 0, 7102]
0
7263130176
{O}
-
1 [9992409 : -107110351 : 1225043] 12.7971812580015
- 7102
7103 [0, 0, 0, 0, 7103]
0
21795527088
{O}
-
1 [-175623 : 155854 : 9261] 9.59271544051242
- 7103
7104 [0, 0, 0, 0, 7104]
0
5322672
{O}
-
1 [-9460 : 109312 : 1331] 6.84473391694478
- 7104
7105 [0, 0, 0, 0, 7105]
0
111264300
{O}
-
2 [-6 : 83 : 1],
[219 : 3242 : 1]
2.96871631586219,
5.44314499354450
[-6, 83], [219, 3242] 7105
7106 [0, 0, 0, 0, 7106]
0
21813941952
{O}
-
0 - - - 7106
7107 [0, 0, 0, 0, 7107]
0
21820081968
{O}
-
2 [-11 : 76 : 1],
[-1928161821915 : -8267156713826 : 131483989125]
3.45456028408170,
20.5441922705179
[-11, 76] 7107
7108 [0, 0, 0, 0, 7108]
0
341034732
{O}
-
1 [12 : 94 : 1] 2.56958828505908
[12, 94] 7108
7109 [0, 0, 0, 0, 7109]
0
1819363716
{O}
-
2 [434 : 3267 : 8],
[-1722 : 17551 : 216]
5.05305848223532,
7.02281503559432
- 7109
7110 [0, 0, 0, 0, 7110]
0
2426500800
{O}
-
0 - - - 7110
7111 [0, 0, 0, 0, 7111]
0
7281550224
{O}
-
1 [145 : 10542 : 125] 6.08871160286874
- 7111
7112 [0, 0, 0, 0, 7112]
0
1365674688
{O}
-
1 [3371758 : 24289208 : 68921] 10.8718658488282
- 7112
7113 [0, 0, 0, 0, 7113]
0
5464235052
{O}
-
0 - - - 7113
7114 [0, 0, 0, 0, 7114]
0
21863086272
{O}
-
1 [62205 : 439823 : 1331] 8.90000783557909
- 7114
7115 [0, 0, 0, 0, 7115]
0
21869233200
{O}
-
2 [-19 : 16 : 1],
[4467 : 57502 : 27]
3.50440364334588,
7.37755158813003
[-19, 16] 7115
7116 [0, 0, 0, 0, 7116]
0
1367211312
{O}
-
0 - - - 7116
7117 [0, 0, 0, 0, 7117]
0
5470382412
{O}
-
1 [1822723002018318711 : 101519481964859018380 : 587575041999093] 30.7150017770880
- 7117
7118 [0, 0, 0, 0, 7118]
0
7295893056
{O}
-
1 [-552933239182142 : 7931924738608779 : 95331258954568] 24.3402743583558
- 7118
7119 [0, 0, 0, 0, 7119]
0
2432647728
{O}
-
1 [-31956 : 48707 : 1728] 8.47101693514116
- 7119
7120 [0, 0, 0, 0, 7120]
0
1782225
{O}
-
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1 [45283 : -521116 : 343] 8.86512108882487
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1 [1747815 : -39775681 : 3375] 11.6888081208348
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1 [1330283791036 : -30167444687115 : 343588336064] 20.6086845234796
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1 [18177315 : 113825729 : 912673] 12.7873709965706
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1 [-12 : 77 : 1] 3.02124373755163
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1 [-76839242 : -247097565 : 4574296] 13.1865136014714
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1 [10276 : 130331 : 64] 7.92701237936300
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1 [36870180315 : -397310410628 : 4370722875] 18.2762555453775
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1 [-1244 : 1147 : 64] 6.30155263850770
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1 [49 : 354 : 1] 3.59625989597183
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1 [3437 : 31949 : 343] 7.38454244432382
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1 [1458 : 55672 : 1] 6.59999363433865
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1 [-13 : 74 : 1] 3.48960652567374
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1 [-416109 : 11521664 : 132651] 11.3068720857944
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1 [14770 : -1795026 : 1] 8.35872144894483
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1 [356010 : -6717849 : 1000] 9.78182197966517
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1 [1824540 : 19656827 : 216000] 11.6668896761490
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1 [975 : 6319 : 27] 6.13315584320869
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1 [62 : 496 : 1] 1.34380435741884
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1 [55380 : 716647 : 8000] 9.45849928451660
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1 [-16 : 60 : 1] 1.57276921408843
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1 [-4377981 : -23985883 : 328509] 11.9606969497841
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1 [-5134315864313014006109249850 : -9399698535378883780318393459 : 274715689465968810196952088] 44.1106143130559
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1 [10340253 : 70021540 : 250047] 12.3117278551794
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1 [15266 : -666873 : 8] 8.39729536273578
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1 [1723797039615553388020 : 11412086848225651406483 : 44568655128369735488] 34.1430150387078
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1 [265824254 : -1654598261 : 8741816] 14.4893308321539
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1 [42 : 286 : 1] 3.34304283698520
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1 [772720599 : 6848100962 : 72511713] 15.5671809133711
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2 [5173 : 36203 : 343],
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1 [-158 : 23 : 8] 4.91906558980782
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1 [23 : 141 : 1] 3.14639225825497
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1 [69619 : 516898 : 1331] 8.99053745996677
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2 [3 : 88 : 1],
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2 [-17 : 53 : 1],
[2 : 703 : 8]
3.51558930898601,
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1 [72072 : 856279 : 512] 9.19267574977256
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1 [-10 : 82 : 1] 2.78237568784134
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7725 [0, 0, 0, 0, 7725]
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1 [45357165 : 1263729430 : 14348907] 13.3636024260249
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7726 [0, 0, 0, 0, 7726]
0
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1 [540219862823265 : 3436775938786517 : 16054501372083] 24.1642620260515
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1 [60 : 473 : 1] 3.83860163727440
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1 [-158 : 41 : 8] 4.91974305784314
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2 [-15 : 66 : 1],
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0 - - - 7732
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2 [47 : 334 : 1],
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4.11591917885549,
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1 [613045011541628811546197195 : -18575796342417298815568579787 : 667707948782576614012625] 43.4076829797676
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1 [9 : 92 : 1] 3.48533408535125
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2 [2 : 88 : 1],
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1 [-8 : 85 : 1] 3.00452807357955
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1 [-7 : 86 : 1] 3.46249616853930
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1 [89288 : -1179971 : 512] 9.39012183346341
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1 [65351 : 405469 : 2197] 8.95191302415698
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[0, 88]
1 [48 : 344 : 1] 2.28295031240535
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1 [-1 : 88 : 1] 3.44741440283801
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1 [-636 : 5265 : 64] 4.60090396821005
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1 [35 : 225 : 1] 1.75866809100548
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1 [-2 : 88 : 1] 2.75567086910931
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2 [7 : 90 : 1],
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1 [-19 : 30 : 1] 2.98039101544136
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1 [322 : 2161 : 8] 5.39424814764911
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1 [39 : 259 : 1] 3.98694422481822
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1 [691119 : 4447387 : 19683] 10.5113047440526
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1 [-14270227626677099 : 639523637330716 : 720600368648077] 26.3425854464545
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1 [-18 : 44 : 1] 2.83050584899977
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1 [14493 : 335789 : 27] 8.50550880745851
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1 [-3 : 88 : 1] 3.45138979819977
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1 [-67292874 : -655059925 : 7762392] 14.0476613293298
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1 [-141 : 2359 : 27] 5.65478492318208
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1 [-4700435639596 : 24077020725467 : 337735169344] 21.1961019411692
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1 [2933101718283864545520205103 : 42750794934623382335460952940 : 13818031137675318801014677] 44.0073685178748
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1 [25 : 153 : 1] 3.18522953717214
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1 [28497 : 261505 : 343] 8.45988230578658
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1 [-158 : 73 : 8] 4.92187861986530
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1 [2916204193 : 80053692469 : 3869893] 16.7534827806631
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1 [638670 : -3915413 : 27000] 10.5121624537118
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1 [1973389727183 : -13701530784468 : 44625548323] 19.8715966228696
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1 [-172776 : 1056943 : 13824] 9.84788218165076
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1 [-42 : 2384 : 27] 4.95437436939941
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1 [1590 : 63401 : 1] 6.91706284594206
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1 [360465 : 3737179 : 3375] 10.2020569999782
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1 [13 : 100 : 1] 3.53291941438150
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1 [-97615 : 3785209 : 42875] 10.5622436283742
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1 [4106698871080053588 : -40854275224746456647 : 41838299749572416] 30.2268712640044
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1 [-153825 : -526042 : 9261] 9.60555006933072
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1 [-62232347030 : -10849728322399 : 122763473000] 20.4723100934235
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1 [-12 : 78 : 1] 1.83297737709265
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1 [875 : 25883 : 1] 6.78817321350602
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1 [-6797 : 1982 : 343] 7.42887685986944
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1 [3518383677354 : 26226107962984 : 257623674147] 20.3656266499819
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1 [-16 : 61 : 1] 3.05096108350326
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1 [-19 : 31 : 1] 3.53198931202295
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1 [27121501978201395268314274488838627537773762287897 : 186398152584623305624837551485596770028144776655756 : 1720094998106353355821008525938727950159777043481] 77.6177737686380
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1 [40 : 268 : 1] 2.61838028535301
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1 [132624 : 837137 : 4096] 9.41691823998468
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1 [-113175069 : -239728625 : 6128487] 13.9478594937441
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2 [-17 : 54 : 1],
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1 [-20559 : 85689 : 1331] 6.84116837802895
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1 [536430 : 3380221 : 27000] 10.4444139555795
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1 [158 : 1988 : 1] 4.44699230753110
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1 [643870575 : 4797177343 : 47045881] 15.3210411555958
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2 [-146993 : -948062 : 12167],
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1 [3097234264098894738 : -30535749157022943835 : 32143478906566664] 30.0351675617704
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1 [2963586248 : -21460421109 : 60236288] 15.5437132270249
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1 [-62 : 687 : 8] 4.30678854257246
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2 [44 : 305 : 1],
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1 [-19766274393761402390 : -133405255543213553857 : 1688942815501969000] 31.4695896411170
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1 [2100 : 18674 : 27] 5.78583536859767
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1 [-52514 : -79421 : 2744] 8.81199504296590
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1 [22 : 136 : 1] 2.98756190639869
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1 [11144 : 69001 : 512] 7.83540962538724
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1 [-34782 : 2639558 : 29791] 9.62694687124212
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1 [-19510 : 20657 : 1000] 8.14166334829619
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1 [17 : 113 : 1] 3.59405402951848
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2 [-18 : 45 : 1],
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[-18, 45], [4, 89], [18, 117], [11734, 1271069] 7857
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1 [98935834696473313 : 634850640256913235 : 2857907585641031] 27.0890226106156
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1 [3260054183 : 34699206320 : 28934443] 16.2873344353142
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2 [-15 : 67 : 1],
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1 [14 : 103 : 1] 3.08662602160638
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0 - - - 7866
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1 [963255789 : -6714478414 : 62570773] 15.5370515207972
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1 [193115909 : 7824081133 : 117649] 15.1944332047584
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2 [-5 : 88 : 1],
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1 [-58418282874 : -77482005031 : 3023464536 18.0890271906877
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1 [27 : 166 : 1] 3.77372646622816
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1 [-110 : 581 : 8] 4.88816605463103
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1 [-5033 : 23554 : 343] 7.39913079476439
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2 [20 : 126 : 1],
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2 [82 : 757 : 8],
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1 [129 : 2413 : 27] 5.65991495312985
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1 [53860940348930287848030 : -814795942772296337569093 : 235510377648526323000] 36.7581914448680
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1 [-789206 : 17248189 : 195112] 11.5793845412237
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1 [6 : 90 : 1] 1.31113855216420
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2 [11 : 96 : 1],
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1 [1582367574451811003262 : 95129001987175387668583 : 1071015530200021913976] 35.7358568357804
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1 [1568685972 : -80702164423 : 592704] 16.7473129340398
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1 [91644 : 1498300 : 343] 8.13933608408043
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1 [-10 : 83 : 1] 3.02011901734975
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0 - - - 7890
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2 [69 : 580 : 1],
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2 [-11 : 81 : 1],
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1 [2289 : 21212 : 27] 6.78315145261589
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1
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7895 [0, 0, 0, 0, 7895]
0
26927002800
{O}
-
1 [9884697762439760510030636 : 62889879477731982306645581 : 511811018456086713716416] 40.0289380540261
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7896 [0, 0, 0, 0, 7896]
0
1683364032
{O}
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1 [-49970 : 594376 : 6859] 8.66588530840480
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7897 [0, 0, 0, 0, 7897]
0
6735161772
{O}
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0 - - - 7897
7898 [0, 0, 0, 0, 7898]
0
26947470528
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-
0 - - - 7898
7899 [0, 0, 0, 0, 7899]
0
26954294832
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0 - - - 7899
7900 [0, 0, 0, 0, 7900]
0
67402800
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1 [21 : 131 : 1] 3.66399797432040
[21, 131] 7900
7901 [0, 0, 0, 0, 7901]
0
2247328836
{O}
-
0 - - - 7901
7902 [0, 0, 0, 0, 7902]
0
2997196992
{O}
-
1 [-40158528 : 621862003 : 7077888] 13.9793198268598
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7903 [0, 0, 0, 0, 7903]
0
8993866896
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1 [-40158528 : 621862003 : 7077888] 13.9793198268598
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7904 [0, 0, 0, 0, 7904]
0
105423552
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1 [98808060 : 606644693 : 3796416] 13.8560238775522
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7905 [0, 0, 0, 0, 7905]
0
6748814700
{O}
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2 [57 : 2402 : 27],
[-2470 : 1721 : 125]
5.65289841071454,
6.29680089059747
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7906 [0, 0, 0, 0, 7906]
0
27002089152
{O}
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1 [1850065125 : 11371711321 : 69426531] 15.8050720579633
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7907 [0, 0, 0, 0, 7907]
0
27008920368
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0 - - - 7907
7908 [0, 0, 0, 0, 7908]
0
422121132
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1 [-8 : 86 : 1] 2.54946228289515
[-8, 86] 7908
7909 [0, 0, 0, 0, 7909]
0
6755646348
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1 [6260185136940 : 53439152694467 : 552175273664] 21.5409023498598
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7910 [0, 0, 0, 0, 7910]
0
9009806400
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1 [6054 : 37367 : 216] 7.37707374817058
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7911 [0, 0, 0, 0, 7911]
0
37086768
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2 [37 : 242 : 1],
[13840 : 365201 : 4096]
3.95493251728955,
9.00410042597132
[37, 242] 7911
7912 [0, 0, 0, 0, 7912]
0
563397696
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1 [-7 : 87 : 1] 2.92035299935590
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7913 [0, 0, 0, 0, 7913]
0
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2 [2 : 89 : 1],
[131 : 1502 : 1]
2.99372004593858,
4.96917361816237
[2, 89], [131, 1502] 7913
7914 [0, 0, 0, 0, 7914]
0
27056763072
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1 [-1470474612422413568659623392920 : -30404319574727288285504480801819 : 343479585642835085219401759232] 48.7997956674370
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7915 [0, 0, 0, 0, 7915]
0
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1 [-2878344 : -10407571 : 175616] 11.5698912979038
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7916 [0, 0, 0, 0, 7916]
0
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0 - - - 7916
7917 [0, 0, 0, 0, 7917]
0
6769320012
{O}
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1 [157848 : 1340695 : 13824] 9.87348442818228
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7918 [0, 0, 0, 0, 7918]
0
27084120768
{O}
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0 - - - 7918
7919 [0, 0, 0, 0, 7919]
0
9030320784
{O}
-
1 [2235 : 20476 : 27] 6.76303498249461
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7920 [0, 0, 0, 0, 7920]
0
11761200
{O}
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2 [9 : 93 : 1],
[1 : 89 : 1]
2.76016133418060,
3.45487776272904
[1, 89], [9, 93] 7920
7921 [0, 0, 0, 0, 7921]
0
285156
Z/3Z
[0, 89]
1 [84 : 775 : 1] 4.11616465047444
[0, 89], [84, 775] 7921
7922 [0, 0, 0, 0, 7922]
0
27111492288
{O}
-
1 [-1 : 89 : 1] 3.45494084536420
[-1, 89] 7922
7923 [0, 0, 0, 0, 7923]
0
27118337328
{O}
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1 [1653502158721328898467209 : 10444438274932791241385798 : 83056675091979536846137] 38.8278007380180
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7924 [0, 0, 0, 0, 7924]
0
423830988
{O}
-
0 - - - 7924
7925 [0, 0, 0, 0, 7925]
0
271320300
{O}
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2 [-158 : 119 : 8],
[1424 : 364637 : 4096]
4.92698859669566,
8.99991529114032
- 7925
7926 [0, 0, 0, 0, 7926]
0
27138877632
{O}
-
0 - - - 7926
7927 [0, 0, 0, 0, 7927]
0
27145726128
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0 - - - 7927
7928 [0, 0, 0, 0, 7928]
0
565678656
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1 [-14 : 72 : 1] 2.26296000164165
[-14, 72] 7928
7929 [0, 0, 0, 0, 7929]
0
754428492
{O}
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2 [-2 : 89 : 1],
[328 : 5941 : 1]
2.99422373016659,
5.36833418559501
[-2, 89], [328, 5941] 7929
7930 [0, 0, 0, 0, 7930]
0
9055425600
{O}
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1 [8623605773311 : 84804328438389 : 89975616641] 20.9576123038330
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7931 [0, 0, 0, 0, 7931]
0
27173128752
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-
1 [2005 : 13726 : 125] 6.80000965600045
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7932 [0, 0, 0, 0, 7932]
0
1698748848
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1 [-537 : 217 : 27] 5.73906835531481
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7933 [0, 0, 0, 0, 7933]
0
6796708812
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-
0 - - - 7933
7934 [0, 0, 0, 0, 7934]
0
27193689792
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0 - - - 7934
7935 [0, 0, 0, 0, 7935]
0
51418800
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-
1 [39068423604 : -373486231817 : 432081216] 17.8974256781929
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7936 [0, 0, 0, 0, 7936]
0
415152
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0 - - - 7936
7937 [0, 0, 0, 0, 7937]
0
2267854884
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0 - - - 7937
7938 [0, 0, 0, 0, 7938]
0
762048
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2 [7 : 91 : 1],
[-17 : 55 : 1]
2.18024380680297,
3.52381228623015
[-17, 55], [7, 91] 7938
7939 [0, 0, 0, 0, 7939]
0
9075991824
{O}
-
1 [53 : 396 : 1] 3.65773008481418
[53, 396] 7939
7940 [0, 0, 0, 0, 7940]
0
425544300
{O}
-
2 [-16 : 62 : 1],
[7945 : 64313 : 125]
2.59355963150267,
7.56713533015835
[-16, 62] 7940
7941 [0, 0, 0, 0, 7941]
0
6810423948
{O}
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0 - - - 7941
7942 [0, 0, 0, 0, 7942]
0
75480768
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1 [-45000372332559 : -160946750598311 : 2731670512739] 22.6113439612017
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7943 [0, 0, 0, 0, 7943]
0
161274672
{O}
-
0 - - - 7943
7944 [0, 0, 0, 0, 7944]
0
1703892672
{O}
-
0 - - - 7944
7945 [0, 0, 0, 0, 7945]
0
6817286700
{O}
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1 [-438 : 1637 : 27] 5.25442408495313
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7946 [0, 0, 0, 0, 7946]
0
9092003904
{O}
-
1 [260 : 5729 : 64] 6.23522179032865
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7947 [0, 0, 0, 0, 7947]
0
3031430832
{O}
-
0 - - - 7947
7948 [0, 0, 0, 0, 7948]
0
568536336
{O}
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3 [-18 : 46 : 1],
[-19 : 33 : 1],
[-3 : 89 : 1]
2.83727428839798,
2.98743858899049,
3.45885074294891
[-19, 33], [-18, 46], [-3, 89], [102, 1034], [206, 2958], [6278, 497430] 7948
7949 [0, 0, 0, 0, 7949]
0
6824152908
{O}
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0 - - - 7949
7950 [0, 0, 0, 0, 7950]
0
1092139200
{O}
-
0 - - - 7950
7951 [0, 0, 0, 0, 7951]
0
27310349232
{O}
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1 [-1450598790 : -7843095449 : 106496424] 15.8265337722358
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7952 [0, 0, 0, 0, 7952]
0
6669243
{O}
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2 [8 : 92 : 1],
[696 : 4276 : 27]
2.09909287453830,
4.56367588200445
[8, 92] 7952
7953 [0, 0, 0, 0, 7953]
0
6831022572
{O}
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1 [-670 : 11039 : 125] 6.22216033601936
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7954 [0, 0, 0, 0, 7954]
0
27330962112
{O}
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4 [-9 : 85 : 1],
[210 : 1291 : 8],
[255 : 4073 : 1],
[1743 : 72769 : 1]
3.48001823474526,
5.14783094112277,
5.58946841543201,
7.47040671890620
[-9, 85], [255, 4073], [1743, 72769] 7954
7955 [0, 0, 0, 0, 7955]
0
9112611600
{O}
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1 [190526 : 1606347 : 2744] 9.14854835264724
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7956 [0, 0, 0, 0, 7956]
0
47473452
{O}
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0 - - - 7956
7957 [0, 0, 0, 0, 7957]
0
2279298564
{O}
-
0 - - - 7957
7958 [0, 0, 0, 0, 7958]
0
27358458048
{O}
-
1 [27819 : 183743 : 729] 8.36998925894589
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7959 [0, 0, 0, 0, 7959]
0
27365334192
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1 [1334353072328612520 : -8349329421784567369 : 43438995090749952] 29.3840574908641
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7960 [0, 0, 0, 0, 7960]
0
1710763200
{O}
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1 [-6 : 88 : 1] 2.77470899440253
[-6, 88] 7960
7961 [0, 0, 0, 0, 7961]
0
6844772268
{O}
-
1 [-72065 : 2172566 : 24389] 10.1938924700592
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7962 [0, 0, 0, 0, 7962]
0
27385967808
{O}
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0 - - - 7962
7963 [0, 0, 0, 0, 7963]
0
27392847408
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0 - - - 7963
7964 [0, 0, 0, 0, 7964]
0
570827664
{O}
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1 [-2786 : 29562 : 343] 6.12582830592537
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7965 [0, 0, 0, 0, 7965]
0
9398700
{O}
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1 [-14353824 : -53777591 : 884736] 12.6487591094794
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7966 [0, 0, 0, 0, 7966]
0
9137830464
{O}
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1 [-1328933 : 1945803 : 68921] 10.4170412840135
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7967 [0, 0, 0, 0, 7967]
0
27420374448
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1 [1316 : 8261 : 64] 6.43056934714973
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7968 [0, 0, 0, 0, 7968]
0
107137728
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1 [1298 : 16549 : 8] 6.55137129016165
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7969 [0, 0, 0, 0, 7969]
0
6858535788
{O}
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2 [-12 : 79 : 1],
[30 : 187 : 1]
3.03368343109510,
3.36893603891227
[-12, 79], [30, 187] 7969
7970 [0, 0, 0, 0, 7970]
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27441028800
{O}
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0 - - - 7970
7971 [0, 0, 0, 0, 7971]
0
27447915312
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2 [19210 : 122719 : 1000],
[-14736138 : -1329031583 : 14886936]
8.23962823463657,
14.4676503605139
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7972 [0, 0, 0, 0, 7972]
0
428981292
{O}
-
1 [1668 : 10255 : 64] 6.53104169514653
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7973 [0, 0, 0, 0, 7973]
0
2288474244
{O}
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2 [-13 : 76 : 1],
[127 : 1434 : 1]
3.50145022844802,
4.39208992430500
[-13, 76], [127, 1434] 7973
7974 [0, 0, 0, 0, 7974]
0
3052064448
{O}
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2 [51 : 375 : 1],
[-15 : 2411 : 27]
3.44616939391340,
5.65408697820844
[51, 375] 7974
7975 [0, 0, 0, 0, 7975]
0
366339600
{O}
-
2 [5 : 90 : 1],
[978 : 10837 : 8]
1.84541371486599,
6.29339284015571
[5, 90] 7975
7976 [0, 0, 0, 0, 7976]
0
1717647552
{O}
-
1 [-251112 : -615473 : 13824] 9.88860452289820
- 7976
7977 [0, 0, 0, 0, 7977]
0
6872313132
{O}
-
2 [28 : 173 : 1],
[-1120 : 10649 : 125]
3.33229856492799,
6.23753259984814
[28, 173] 7977
7978 [0, 0, 0, 0, 7978]
0
27496145088
{O}
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1 [7278372 : 91002529 : 46656] 12.2959311464337
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7979 [0, 0, 0, 0, 7979]
0
27503038512
{O}
-
0 - - - 7979
7980 [0, 0, 0, 0, 7980]
0
1719370800
{O}
-
0 - - - 7980
7981 [0, 0, 0, 0, 7981]
0
6879206988
{O}
-
1 [1949982038966181 : 19430406104800054 : 206917948969523] 25.4743741564185
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7982 [0, 0, 0, 0, 7982]
0
9174574656
{O}
-
1 [272 : 365945 : 4096] 9.00225869406152
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7983 [0, 0, 0, 0, 7983]
0
3058957872
{O}
-
1 [692952 : 5059205 : 13824] 10.5219761111942
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7984 [0, 0, 0, 0, 7984]
0
35856144
{O}
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1 [-1155 : 10603 : 125] 6.70121941211398
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7985 [0, 0, 0, 0, 7985]
0
6886104300
{O}
-
2 [-4 : 89 : 1],
[25068 : 181547 : 1728]
3.00060717220331,
8.52976233954290
[-4, 89] 7985
7986 [0, 0, 0, 0, 7986]
0
1881792
{O}
-
1 [-43901 : 5929 : 2197] 6.27641899558265
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7987 [0, 0, 0, 0, 7987]
0
562412592
{O}
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1 [-126 : 511 : 8] 3.60704254499921
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7988 [0, 0, 0, 0, 7988]
0
430704972
{O}
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1 [-2495 : 749 : 125] 6.76322134192398
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7989 [0, 0, 0, 0, 7989]
0
6893005068
{O}
-
1 [507092914979551912830 : 4182191970392343993047 : 7660448551883480184] 33.3696710777371
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7990 [0, 0, 0, 0, 7990]
0
27578923200
{O}
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0 - - - 7990
7991 [0, 0, 0, 0, 7991]
0
9195275664
{O}
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1 [-361515 : -3000898 : 35937] 10.4795973247589
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0 - - - 7992
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0 - - - 7993
7994 [0, 0, 0, 0, 7994]
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1 [72590329322560276470 : 910120360418927681309 : 9940011454811443032] 32.6439056413676
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7995 [0, 0, 0, 0, 7995]
0
27613450800
{O}
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1 [121 : 1334 : 1] 4.89796088803103
[121, 1334] 7995
7996 [0, 0, 0, 0, 7996]
0
1726272432
{O}
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1 [-30 : 713 : 8] 4.84882337728291
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7997 [0, 0, 0, 0, 7997]
0
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{O}
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3 [23 : 142 : 1],
[-158 : 137 : 8],
[201 : 2476 : 27]
3.70263278736779,
4.92962150292754,
5.68019353543819
[23, 142] 7997
7998 [0, 0, 0, 0, 7998]
0
27634177728
{O}
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0 - - - 7998
7999 [0, 0, 0, 0, 7999]
0
27641088432
{O}
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2 [-15 : 68 : 1],
[6762542910 : -47999632847 : 449455096]
3.51399411974951,
16.8506153620314
[-15, 68] 7999
8000 [0, 0, 0, 0, 8000]
0
900
Z/2Z
[-20, 0]
0 - - - 8000

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Last Update: 2022.09.19
H.Nakao

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