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$\pi/3$-congruent numbers and $2\pi/3$-congruent numbers

[2007.12.05](π/3)-合同数と(2π/3)-合同数(n \in [101..500])

θ-合同数の定義については、こちらを参照のこと。

以下では、特に、θ=(π/3),(2π/3)の場合について、θ-合同数を取り上げる。

■(π/3)-合同数
自然数nについて、三辺a,b,cの長さが有理数の三角形で、∠C=π/3で、面積がn√(3)である(つまり、c^2=a^2-ab+b^2, ab=4nである)ものが存在するとき、nは(π/3)-合同数であるという。

■(2π/3)-合同数
同様に、自然数nについて、三辺a,b,cの長さが有理数の三角形で、∠C=2π/3で、面積がn√(3)である(つまり、c^2=a^2+ab+b^2, ab=4nである)ものが存在するとき、nは(2π/3)-合同数であるという。

■101以上500以下のsquare-freeな自然数nについて、nが(π/3)-合同数あるいは(2π/3)-合同数かどうかを調べてみる。
各nに対して、Mordell-Weil群En,π/3(Q),En,2π/3(Q)のrankと生成元を調べるために、Cremonaのmwrankとheegner.gpを使った。

これらのMordell-Weil群で生成元が決定できなかったのは、E353,2π/3(Q),E358,2π/3(Q),E499,2π/3(Q)の３つである。

  En,π/3: y2=x(x+3n)(x-n)
Y2Z=X(X+3nZ)(X-nZ)
Δ(En,π/3)=2304n6=28・32n6
j(En,π/3)=35152/9=24・132/32 		En,2π/3: y2=x(x+n)(x-3n)
Y2Z=X(X+nZ)(X-3nZ)
Δ(En,2π/3)=2304n6=28・32n6
j(En,2π/3)=35152/9=24・132/32 		 
n [a1,a2,a3,a4,a6]
conductor of En,π/3		 rank(
En,π/3)
 En,π/3(Q)/En,π/3(Q)torsの生成元[X:Y:Z]
 En,π/3(Q)/
En,π/3(Q)tors
の生成元の高さ
 nはπ/3-合同数か？
π/3-有理三角形(a,b,c):
c2=a2+b2-ab, ab=4n		 [a1,a2,a3,a4,a6]
conductor of En,2π/3		 rank(
En,2π/3)
 En,2π/3(Q)/En,2π/3(Q)torsの生成元[X:Y:Z]
 En,2π/3(Q)/
En,2π/3(Q)tors
の生成元の高さ
 nは2π/3-合同数か？
2π/3-有理三角形(a,b,c):
c2=a2+b2+ab, ab=4n		 n
101 [0, 202, 0, -30603, 0]
244824		 0 - - π/3-非合同数 [0, -202, 0, -30603, 0]
489648		 1 [-75336744145 : -1283476601224 : 1349232625] 18.5697959918429 2π/3-合同数
(155441032/9123985, 921522485/38860258, 12568580455180819/354560411088130)
 101
102 [0, 204, 0, -31212, 0]
166464		 1 [-150 : 11440 : 27] 6.00741047049689 π/3-合同数
(765/143, 1144/15, 158167/2145)
 [0, -204, 0, -31212, 0]
166464		 0 - - 2π/3-非合同数
 102
103 [0, 206, 0, -31827, 0]
509232		 0 - - π/3-非合同数
 [0, -206, 0, -31827, 0]
254616		 0 - - 2π/3-非合同数
 103
105 [0, 210, 0, -33075, 0]
88200		 2 [-245 : 2450 : 1],
[-300 : 1350 : 1]
 1.3700523613161,
3.16718363163946
 π/3-合同数
(10, 42, 38),
(9/2, 280/3, 547/6)
 [0, -210, 0, -33075, 0]
176400		 1 [-5 : 400 : 1] 2.90828819918262 2π/3-合同数
(21/4, 80, 331/4)
 105
106 [0, 212, 0, -33708, 0]
539328		 1 [-294 : 1680 : 1] 3.93643374512696 π/3-合同数
(40/7, 371/5, 2503/35)
 [0, -212, 0, -33708, 0]
539328		 1 [16985804490 : 218317746528 : 37595375] 15.9102796537282 2π/3-合同数
(12516784/973845, 51613785/1564598, 62404554342307/1523675939310)
 106
107 [0, 214, 0, -34347, 0]
549552		 1 [-167723303270746680000 : -37627496065301391477800 : 244728087752194965729] 31.8497473141678 π/3-合同数
(12525030021600/6565184241187, 702474713807009/3131257505400, 4592391974868739970768759683/20557282429550597554909800) 		[0, -214, 0, -34347, 0]
274776		 0 - - 2π/3-非合同数
 107
109 [0, 218, 0, -35643, 0]
285144		 1 [75603817545 : 1460631415544 : 268336125] 13.5511421925474 π/3-合同数
(12922168/668865, 668865/29638, 418912222009/19823820870)
 [0, -218, 0, -35643, 0]
570288		 0 - - 2π/3-非合同数
 109
110 [0, 220, 0, -36300, 0]
580800		 0 - - π/3-非合同数
 [0, -220, 0, -36300, 0]
580800		 0 - - 2π/3-非合同数
 110
111 [0, 222, 0, -36963, 0]
197136		 0 - - π/3-非合同数
 [0, -222, 0, -36963, 0]
98568		 1 [-108 : 378 : 1] 3.75125020417445 2π/3-合同数
(7/2, 888/7, 1801/14)
 111
113 [0, 226, 0, -38307, 0]
306456		 1 [-3075061361 : 111707717970 : 90518849] 16.7376663198528 π/3-合同数
(265557458/21342705, 42685410/1175033, 801907266179714/2507838268426)
 [0, -226, 0, -38307, 0]
612912
 1 [-511225 : -15439424 : 15625] 11.0662416133054 2π/3-合同数
(403975/26992, 107968/3575, 3845458369/96496400)
 113
114 [0, 228, 0, -38988, 0]
207936		 1 [-48 : 1512 : 1] 3.64718204232873 π/3-合同数
(304/21, 63/2, 1147/42)
 [0, -228, 0, -38988, 0]
207936		 2 [-108 : 540 : 1],
[-50 : 1120 : 1]
 3.08143045169487,
3.96614716700157
 2π/3-合同数
(5, 456/5, 469/5),
(285/14, 112/5, 2593/70)
 114
115 [0, 230, 0, -39675, 0]
634800		 0 - - π/3-非合同数
 [0, -230, 0, -39675, 0]
317400		 1 [-8454225 : -451131200 : 658503] 8.80567099905435 2π/3-合同数
(5655/656, 60352/1131, 43145827/741936)
 115
118 [0, 236, 0, -41772, 0]
668352		 1 [-3750 : 74800 : 27] 6.49089765625316 π/3-合同数
(1496/75, 4425/187, 309127/14025)
 [0, -236, 0, -41772, 0]
668352		 1 [-701145415130333184 : -21047786309233376000 : 19684968365610729] 25.6085143191685 2π/3-合同数
(226582912512/14410647125, 850228180375/28322864064, 16429634942314858403107/408150799595647416000)
 118
119 [0, 238, 0, -42483, 0]
679728		 1 [1344 : 52920 : 1] 4.80939334924011 π/3-合同数
(544/45, 315/8, 12577/360)
 [0, -238, 0, -42483, 0]
339864		 1 [-21 : 882 : 1] 1.99453582559505 2π/3-合同数
(34/3, 42, 146/3)
 119
122 [0, 244, 0, -44652, 0]
714432		 0 - - π/3-非合同数
 [0, -244, 0, -44652, 0]
714432		 0 - - 2π/3-非合同数
 122
123 [0, 246, 0, -45387, 0]
242064		 0 - - π/3-非合同数
 [0, -246, 0, -45387, 0]
121032		 0 - - 2π/3-非合同数
 123
127 [0, 254, 0, -48387, 0]
774192		 0 - - π/3-非合同数
 [0, -254, 0, -48387, 0]
387096		 0 - - 2π/3-非合同数
 127
129 [0, 258, 0, -49923, 0]
133128		 0 - - π/3-非合同数
 [0, -258, 0, -49923, 0]
266256		 1 [-17462601 : -726878880 : 704969] 8.31427244273147 2π/3-合同数
(3471/280, 48160/1157, 15878347/323960)
 129
130 [0, 260, 0, -50700, 0]
811200		 1 [-270 : 3600 : 1] 2.41572804450893 π/3-合同数
(40/3, 39, 103/3)
 [0, -260, 0, -50700, 0]
811200		 1 [1470 : 50400 : 1] 3.78927599941262 2π/3-合同数
(91/6, 240/7, 1843/42)
 130
131 [0, 262, 0, -51483, 0]
823728		 1 [-13556928 : -62466040 : 35937] 8.21767478434441 π/3-合同数
(8515/1848, 7392/65, 13392259/120120)
 [0, -262, 0, -51483, 0]
411864		 0 - - 2π/3-非合同数
 131
133 [0, 266, 0, -53067, 0]
424536		 1 [2401 : 123480 : 1] 5.2098705764365 π/3-合同数
(931/90, 360/7, 29683/630)
 [0, -266, 0, -53067, 0]
849072		 2 [-108 : 1170 : 1],

 5.00940397820221,
5.71775447133254
 2π/3-合同数
(65/6, 3192/65, 21577/390),
(280/69, 1311/10, 91891/690)
 133
134 [0, 268, 0, -53868, 0]
861888
 2 [-108 : 2772 : 1],
[-5046 : 96976 : 27]
 4.47143705414671,
6.66725490308556
 π/3-合同数
(1608/77, 77/3, 5461/231),
(1672/87, 5829/209, 449527/18183) 		[0, -268, 0, -53868, 0]
861888		 0 - - 2π/3-非合同数
 134
137 [0, 274, 0, -56307, 0]
450456		 1 [-666294190873 : 40797838129050 : 41314084993] 20.9461437441646 π/3-合同数
(13150227494/1469345175, 2938690350/47993531, 4039535377907451314/70519063206062925)
 [0, -274, 0, -56307, 0]
900912		 1 [-4503330090921805 : -1242770043046812896 : 6112927801001125] 29.0097914955206 2π/3-合同数
(3931234969805/1979727602552, 7918910410208/28695145765, 15733992500760067768082209/56808572130223625992280)
 137
138 [0, 276, 0, -57132, 0]
304704		 1 [-86250 : 383525 : 216] 6.60200325256328 π/3-合同数
(667/150, 3600/29, 530593/4350)
 [0, -276, 0, -57132, 0]
304704		 0 - - 2π/3-非合同数
 138
139 [0, 278, 0, -57963, 0]
927408		 0 - - π/3-非合同数
 [0, -278, 0, -57963, 0]
463704		 1 [-38557193956621165816245248 : -1021288947730709818414232160 : 688862941153283713598657] 37.2038955212127 2π/3-合同数
(19798278912965504/943182770103405, 131102405044373295/4949569728241376, 192379810764346379140919563154179/4668348887102658502040319485280)
 		139
141 [0, 282, 0, -59643, 0]
159048		 1 [-1801604 : -54842843 : 21952] 8.51562706421423 π/3-合同数
(12432/671, 31537/1036, 18469873/695156)
 [0, -282, 0, -59643, 0]
318096		 1 [1029717360 : 18041053212 : 1520875] 10.6029767951058 2π/3-合同数
(169247/9660, 115920/3601, 1519153297/34785660)
 141
142 [0, 284, 0, -60492, 0]
967872		 1 [-375590 : -3130461 : 1000] 6.48696103377003 π/3-合同数
(1917/230, 1840/27, 399841/6210)
 [0, -284, 0, -60492, 0]
967872		 1 [268982923544785618 : 10659946547134851840 : 142572066633521] 27.9188162413781 2π/3-合同数
(5951294563277/415234498320, 3321875986560/83821050187, 1685098252492020382767601/34805391723054487185840)
 142
143 [0, 286, 0, -61347, 0]
981552		 1 [-216816600 : -8623445545 : 4741632] 10.9171374411487 π/3-合同数
(63840/4439, 634777/15960, 2471233153/70846440)
 [0, -286, 0, -61347, 0]
490776		 1 [-121 : 1210 : 1] 2.37144374732531 2π/3-合同数
(10, 286/5, 314/5)
 143
145 [0, 290, 0, -63075, 0]
504600		 0 - - π/3-非合同数
 [0, -290, 0, -63075, 0]
1009200		 0 - - 2π/3-非合同数
 145
146 [0, 292, 0, -63948, 0]
1023168
 2 [-432 : 1224 : 1],
[2450 : 127680 : 1]
 4.97757618783247,
6.4875047527328
 π/3-合同数
(17/6, 3504/17, 20881/102),
(2555/228, 1824/35, 379153/7980)
 [0, -292, 0, -63948, 0]
1023168
 2 [-48 : 1512 : 1],
[2166 : 93024 : 1]
 4.34126146318683,
5.77674624501942
 2π/3-合同数
(1168/63, 63/2, 5521/126),
(1387/102, 816/19, 99073/1938)
 146
149 [0, 298, 0, -66603, 0]
532824
 0 - - π/3-非合同数
 [0, -298, 0, -66603, 0]
1065648
 1 [-53711397332037545 : -1957382714435936312 : 1343562819313625] 28.1245878650646 2π/3-合同数
(11470855881805/701388856238, 2805555424952/76985609945, 2527797782809064996412499/53996848906108348086910)
 149
151 [0, 302, 0, -68403, 0]
1094448
 0 - - π/3-非合同数
 [0, -302, 0, -68403, 0]
547224
 0 - - 2π/3-非合同数
 151
154 [0, 308, 0, -71148, 0]
1138368
 1 [-294 : 4704 : 1] 2.21785611851197 π/3-合同数
(16, 77/2, 67/2)
 [0, -308, 0, -71148, 0]
1138368
 1 [-350 : 12495 : 8] 5.07076758271497 2π/3-合同数
(880/51, 357/10, 23857/510)
 154
155 [0, 310, 0, -72075, 0]
1153200
 1 [1515580 : 56410950 : 1331] 10.373310909986 π/3-合同数
(226424/13593, 67965/1826, 801543139/24820818)
 [0, -310, 0, -72075, 0]
576600
 2 [-75 : 1800 : 1],
[1445 : 47600 : 1]
 2.81431588664609,
5.02304453188372
 2π/3-合同数
(24, 155/6, 259/6),
(527/28, 560/17, 21601/476)
 155
157 [0, 314, 0, -73947, 0]
591576
 1 [547978074866104830025 : 50178804018735025237560 : 67815985855450573] 29.1085778348465 π/3-合同数
(1193036630305/173960744694, 109247347667832/1193036630305, 18334564488891467012263417/207541540655078168351670)
 [0, -314, 0, -73947, 0]
1183152
 0 - - 2π/3-非合同数
 157
158 [0, 316, 0, -74892, 0]
1198272
 0 - - π/3-非合同数
 [0, -316, 0, -74892, 0]
1198272
 0 - - 2π/3-非合同数
 158
159 [0, 318, 0, -75843, 0]
404496
 0 - - π/3-非合同数
 [0, -318, 0, -75843, 0]
202248
 1 [-9 : 810 : 1] 3.11289007627097 2π/3-合同数
(106/15, 90, 1406/15)
 159
161 [0, 322, 0, -77763, 0]
622104
 1 [-189 : 4410 : 1] 2.62830698082571 π/3-合同数
(70/3, 138/5, 386/15)
 [0, -322, 0, -77763, 0]
1244208
 1 [-35000 : -956725 : 512] 7.90858471820667 2π/3-合同数
(18400/781, 5467/200, 6890977/156200)
 161
163 [0, 326, 0, -79707, 0]
1275312
 0 - - π/3-非合同数
 [0, -326, 0, -79707, 0]
637656
 1 [-108046673644747785026844347000000 : -2280976505259472495632802475061000 : 1171106656522043847591568968449] 46.7914860393996 2π/3-合同数
(1764631594743831131283/83588135862367741000, 334352543449470964000/10825960703949884241, 40987241367090241146446113352406085811203/904921874162417233591111985122669581000) 		163
165 [0, 330, 0, -81675, 0]
217800
 1 [-11 : 968 : 1] 2.59127074311047 π/3-合同数
(15/2, 88, 169/2)
 [0, -338, 0, -85683, 0]
48
 0 - - 2π/3-非合同数
 165
166 [0, 332, 0, -82668, 0]
1322688
 1 [-7942 : -945744 : 1331] 9.08553586527782 π/3-合同数
(17347/3111, 24888/209, 75678967/650199)
 [0, -336, 0, -84672, 0]
28224
 0 - - 2π/3-非合同数
 166
167 [0, 334, 0, -83667, 0]
1338672
 1 [-203634812125675644373136400 : -971914650645932376190213715 : 420678143611624370368512] 37.4915838048865 π/3-合同数
(45621427732073989/9558566545891440, 38234266183565760/273182201988467, 359395402667367649921201953438913/2611230256859918684177114022480)
 		[0, -334, 0, -83667, 0]
669336
 1 [5733188124712049 : 100293894700410010 : 7630247475703] 20.512166772925 2π/3-合同数
(14381624170/822109433, 1644218866/43058755, 1745768797628184722/35399008658735915)
 167
170 [0, 340, 0, -86700, 0]
1387200
 0 - - π/3-非合同数
 [0, -340, 0, -86700, 0]
1387200
 0 - - 2π/3-非合同数
 170
173 [0, 346, 0, -89787, 0]
718296
 0 - - π/3-非合同数
 [0, -346, 0, -89787, 0]
1436592
 1 [-50625413000455298335525 : -23671878316833783975442232 : 123472700454738760328125] 40.4567026176374 2π/3-合同数
(274679058740664175/185602403773877842, 742409615095511368/1587740223934475, 138011585446475477187441798201624019/294688402130713652698433262402950)
 		173
174 [0, 348, 0, -90828, 0]
484416
 1 [21870 : -42768 : 125] 6.03995804525377 π/3-合同数
(88/45, 3915/11, 175693/495)
 [0, -348, 0, -90828, 0]
484416
 0 - - 2π/3-非合同数
 174
177 [0, 354, 0, -93987, 0]
250632
 0 - - π/3-非合同数
 [0, -354, 0, -93987, 0]
501264
 1 [6811020456332400 : 1367776867515255 : 12826049933312] 20.8148630111135 2π/3-合同数
(63530669/316359120, 3796309440/1076791, 1201031319773584171/340652653183920)
 177
178 [0, 356, 0, -95052, 0]
1520832
 1 [111571922 : 3871588160 : 117649] 13.4496895666275 π/3-合同数
(4653187/226780, 1814240/52283, 358286542321/11856738740)
 [0, -356, 0, -95052, 0]

 1 [-152065888599130018110246 : -2231194853712837843516160 : 1129361758106564188107] 36.7736370438685 2π/3-合同数
(13056059260232320/889828716531117, 79194755771269413/1632007407529040, 83195805071063210238343434750721/1452207056810841274177241137680)
 		178
179 [0, 358, 0, -96123, 0]

 1 [-2158413843983198484672 : -30613301011675641567880 : 5420176679239459857] 29.8770710231045 π/3-合同数
(65275748618405/4602315818232, 18409263272928/364668986695, 75684847158157352801613379/1678321845885033246423240)
 [0, -358, 0, -96123, 0]
768984
 0 - - 2π/3-非合同数
 179
181 [0, 362, 0, -98283, 0]
786264
 1 [28893791411793009 : 722262523839888680 : 61298138750589] 21.7825793116984 π/3-合同数
(62713553320/2508827841, 2508827841/86620930, 5910584213490087481/217317000797312130)
 [0, -362, 0, -98283, 0]
1572528
 0 - - 2π/3-非合同数
 181
182 [0, 364, 0, -99372, 0]
1589952
 0 - - π/3-非合同数
 [0, -364, 0, -99372, 0]
1589952
 0 - - 2π/3-非合同数
 182
183 [0, 366, 0, -100467, 0]
535824
 0 - - π/3-非合同数
 [0, -366, 0, -100467, 0]
267912
 1 [-35100 : -2697570 : 2197] 9.25465140256038 2π/3-合同数
(95160/9991, 9991/130, 106545481/1298830)
 183
185 [0, 370, 0, -102675, 0]
821400
 1 [1875 : 87750 : 1] 4.6229514991971 π/3-合同数
(1850/117, 234/5, 24122/585)
 [0, -370, 0, -102675, 0]
1642800
 1 [-121 : 2288 : 1] 5.17401323792603 2π/3-合同数
(208/11, 2035/52, 29329/572)
 185
186 [0, 372, 0, -103788, 0]
553536
 1 [-108 : 3780 : 1] 3.63686596471338 π/3-合同数
(744/35, 35, 1069/35)
 [0, -372, 0, -103788, 0]
553536
 0 - - 2π/3-非合同数
 186
187 [0, 374, 0, -104907, 0]
1678512
 0 - - π/3-非合同数
 [0, -374, 0, -104907, 0]
839256
 1 [68719427568 : 116659272015 : 122023936] 15.9428638330136 2π/3-1合同数
(1725295/1016304, 69108672/156845, 70371111952387/159402200880)
 187
190 [0, 380, 0, -108300, 0]
1732800
 1 [-300 : 6300 : 1] 3.39266154217931 π/3-合同数
(21, 760/21, 661/21)
 [0, -380, 0, -108300, 0]
1732800
 1 [-1256850 : -44475200 : 19683] 6.49972085339906 2π/3-合同数
(945/44, 6688/189, 413587/8316)
 190
191 [0, 382, 0, -109443, 0]
1751088
 1 [-11043042923451183430920720 : 131384222033427889544983393 : 23385608196635432448000] 35.498564020523 π/3-合同数
(15298197437849183/1285836673102320, 5143346692409280/80095274543713, 6093948713754558407726167288321/102989441350504865478161714160)
 		[0, -382, 0, -109443, 0]
875544
 1 [867 : 16422 : 1] 5.44774584210792 2π/3-合同数
(322/17, 6494/161, 143522/2737)
 191
193 [0, 386, 0, -111747, 0]

 2 [-529 : 4370 : 1],
[-432 : 6300 : 1]
 5.59983654062704,
5.83504361879654
 π/3-合同数
(190/23, 8878/95, 195794/2185),
(175/12, 9264/175, 99457/2100)
 [0, -386, 0, -111747, 0]
1787952
 0 - - 2π/3-非合同数
 193
194 [0, 388, 0, -112908, 0]
1806528
 0 - - π/3-非合同数
 [0, -388, 0, -112908, 0]
1806528
 0 - - 2π/3-非合同数
 194
195 [0, 390, 0, -114075, 0]
608400
 0 - - π/3-非合同数
 [0, -390, 0, -114075, 0]
304200
 0 - - 2π/3-非合同数
 195
197 [0, 394, 0, -116427, 0]
931416
 0 - - π/3-非合同数
 [0, -394, 0, -116427, 0]
1862832
 1 [-148963631951384127943646477135290238838 : -25717351612325786625911335073529279456045 : 38643580322514946189245169318686342632] 57.7014325452058 2π/3-合同数
(3692489507980278120359272/808982342823400714476335, 159369521536209940751837995/923122376995069530089818, 130664787937626409053852342600379337435146431041779/746789703254177895671733358995303069945385457030)
 		197
199 [0, 398, 0, -118803, 0]
1900848
 0 - - π/3-非合同数
 [0, -398, 0, -118803, 0]
950424
 0 - - 2π/3-非合同数
 199
201 [0, 402, 0, -121203, 0]
323208
 0 - - π/3-非合同数
 [0, -402, 0, -121203, 0]
646416
 1 [-364800 : -15578640 : 6859] 10.6840676697421 2π/3-合同数
(407360/21637, 64911/1520, 1795992907/32888240)
 201
202 [0, 404, 0, -122412, 0]
1958592
 1 [2169323933627188 : 50831224641522260 : 5046848440577] 20.6674685256068 π/3-合同数
(7111279205/303488029, 2427904232/70408705, 651687572539563847/21368199104892445)
 [0, -404, 0, -122412, 0]
1958592
 1 [3326631773845851699376 : 740875517604700581748920 : 66524155656382871] 31.3378100101338 2π/3-合同数
(14612736910384/4027733404035, 406801073807535/1826592113798, 1651993758707599799701643107/7357026072291104632374930) 		202
203 [0, 406, 0, -123627, 0]
1978032
 1 [184481292471 : 10764339939080 : 60698457] 17.4058257585756 π/3-合同数
(93330033/6706570, 187783960/3218277, 1139297766988291/21583599979890)
 [0, -406, 0, -123627, 0]
989016
 2 [-175 : 1960 : 1],
[24843 : 484120 : 27]
 3.3491967311127,
6.48124392474777
 2π/3-合同数
(56/5, 145/2, 787/10),
(760/39, 7917/190, 400963/7410)
 203
205 [0, 410, 0, -126075, 0]
1008600
 1 [1805 : 83600 : 1] 5.3227226934303 π/3-合同数
(779/44, 880/19, 33841/836)
 [0, -410, 0, -126075, 0]
2017200
 2 [-169 : 2184 : 1],
[-735 : 47600 : 27]
 5.29988172562899,
5.78795079374818
 2π/3-合同数
(168/13, 2665/42, 38659/546),
(861/68, 1360/21, 102721/1428)
 205
206 [0, 412, 0, -127308, 0]
2036928
 0 - - π/3-非合同数
 [0, -412, 0, -127308, 0]
2036928
 2 [-108 : 2772 : 1],
[-2987 : -3044783 : 24389]
 4.71398626207062,
7.97714050682763
 2π/3-合同数
(77/3, 2472/77, 11581/231),
(232/287, 29561/29, 8487373/8323)
 206
209 [0, 418, 0, -131043, 0]
1048344
 1 [-513 : 6498 : 1] 2.14020106672082 π/3-合同数
(38/3, 66, 182/3)
 [0, -418, 0, -131043, 0]
2096688
 1 [92455 : 1111264 : 125] 6.85782668840094 2π/3-合同数
(2464/205, 3895/56, 875659/11480)
 209
210 [0, 420, 0, -132300, 0]
705600
 1 [-294 : 7056 : 1] 1.87593179815666 π/3-合同数
(24, 35, 31)
 [0, -420, 0, -132300, 0]
705600
 0 - - 2π/3-非合同数
 210
211 [0, 422, 0, -133563, 0]
2137008
 0 - - π/3-非合同数
 [0, -422, 0, -133563, 0]
1068504
 1 [-3330196292840284579237290752 : -334287662745805918476954790800 : 262076413207712055546914657] 41.1126970594195 2π/3-合同数
(401999027921797696/47811577384509225, 10088242828131446475/100499756980449424, 503751725536860757428519388797678979/4805051907995128799422596173936400)
 		211
213 [0, 426, 0, -136107, 0]
362952
 1 [-132659139180 : 17816934027167 : 17173512000] 17.3738833464075 π/3-合同数
(26346960/4153237, 294879827/2195580, 1196831313008401/9118764092460)
 [0, -426, 0, -136107, 0]
725904
 1 [694589733068281200 : 196138108457770915980 : 8664387850819] 25.2426981589465 2π/3-合同数
(32735879760/10849707877, 770329259267/2727989980, 8402854923862763491441/29597894374383072460)
 213
214 [0, 428, 0, -137388, 0]
2198208
 1 [-5222934 : -136237904 : 19683] 11.4952364980922 π/3-合同数
(219032/8397, 898479/27379, 6902088967/229901463)
 [0, -428, 0, -137388, 0]
2198208
 1 [-39033600 : -141051680 : 185193] 8.77729400083383 2π/3-合同数
(8239/2280, 18240/77, 41908003/175560)
 214
215 [0, 430, 0, -138675, 0]
2218800
 1 [-756800 : 7765800 : 1331] 5.98437570666708 π/3-合同数
(903/88, 1760/21, 146323/1848)
 [0, -430, 0, -138675, 0]
1109400
 1 [-735 : 50050 : 27] 5.66133033837233 2π/3-合同数
(1806/143, 1430/21, 225854/3003)
 215
217 [0, 434, 0, -141267, 0]
1130136
 0 - - π/3-非合同数
 [0, -434, 0, -141267, 0]
2260272
 0 - - 2π/3-非合同数
 217
218 [0, 436, 0, -142572, 0]
2281152
 0 - - π/3-非合同数
 [0, -436, 0, -142572, 0]
2281152
 0 - - 2π/3-非合同数
 218
219 [0, 438, 0, -143883, 0]
767376
 0 - - π/3-非合同数
 [0, -438, 0, -143883, 0]
383688
 0 - - 2π/3-非合同数
 219
221 [0, 442, 0, -146523, 0]
1172184
 0 - - π/3-非合同数
 [0, -442, 0, -146523, 0]
2344368
 3 [-204 : 1734 : 1],
[-169 : 2704 : 1],
[-121 : 3080 : 1]
 2.69875611006202,
2.80204150617764,
5.33688201201011
 2π/3-合同数
(17/2, 104, 217/2),
(16, 221/4, 259/4),
(280/11, 2431/70, 40291/770)
 221
222 [0, 444, 0, -147852, 0]
788544
 1 [-10625290 : -690786448 : 274625] 9.37183942810817 π/3-合同数
(9165/671, 198616/3055, 121711639/2049905)
 [0, -444, 0, -147852, 0]
788544
 0 - - 2π/3-非合同数
 222
223 [0, 446, 0, -149187, 0]
2386992
 0 - - π/3-非合同数
 [0, -446, 0, -149187, 0]
1193496
 0 - - 2π/3-非合同数
 223
226 [0, 452, 0, -153228, 0]
2451648
 1 [294350 : 9901760 : 343] 9.52275620723513 π/3-合同数
(114695/4268, 34144/1015, 133506433/4332020)
 [0, -452, 0, -153228, 0]
2451648
 1 [-19359726356850 : -4412294290234880 : 6620314463433] 24.2663985482313 2π/3-合同数
(48175491705/12145712096, 97165696768/426331785, 1190548829138820802753/5178103117983771360)
 226
227 [0, 454, 0, -154587, 0]

 1 [-29331818746608835075697655520499474326800 : 2125558785163742009048438217278322253073380 : 916331598053429278632066512351250223059] 60.4141157865049 π/3-合同数
(-29331818746608835075697655520499474326800, 2125558785163742009048438217278322253073380, 916331598053429278632066512351250223059)
 		[0, -454, 0, -154587, 0]
1236696
 0 - - 2π/3-非合同数
 227
229 [0, 458, 0, -157323, 0]
1258584
 1 [41675512046126194401 : 1225700377110235526120 : 64186901644063869] 26.4808780147967 π/3-合同数
(7939025100520/269937859521, 269937859521/8667057970, 70924360813649984637241/2339567076766223432370)
 [0, -458, 0, -157323, 0]
2517168
 0 - - 2π/3-非合同数
 229
230 [0, 460, 0, -158700, 0]
2539200
 2 [-490 : 8400 : 1],
[-15 : 1575 : 1]
 3.54512945075721,
3.85512771612884
 π/3-合同数
(120/7, 161/3, 997/21),
(184/21, 105, 2119/21)
 [0, -458, 0, -157323, 0]
2517168
 0 - - 2π/3-非合同数
 230
231 [0, 462, 0, -160083, 0]
853776
 0 - - π/3-非合同数
 [0, -462, 0, -160083, 0]
426888
 1 [-189 : 2646 : 1] 1.65134181920559 2π/3-合同数
(14, 66, 74)
 231
233 [0, 466, 0, -162867, 0]
1302936
 1 [-294293990177 : 9183010011450 : 1320139673] 19.4408096689315 π/3-合同数
(8372977558/280328775, 560657550/17967763, 153916279367602754/5036880991280325)
 [0, -466, 0, -162867, 0]
2605872
 1 [-1707556876521534553894346635358870909184 : -44059239566787368164723970797353160749360 : 13220080145207265505917335944688450659] 57.0667296107696 2π/3-合同数
(62013060897755248556923045/2403373948603802862881648, 9613495794415211451526592/266150475956031109686365, 34461314790981829601351995641077613295811789146609/639659120321227981837304896834366025169394329520)
 		233
235 [0, 470, 0, -165675, 0]
2650800
 0 - - π/3-非合同数
 [0, -470, 0, -165675, 0]
1325400
 1 [-1850625 : -273916000 : 250047] 8.1474693025883 2π/3-合同数
(1575/248, 46624/315, 11818627/78120)
 235
237 [0, 474, 0, -168507, 0]
449352
 1 [-46399032225 : 77402659320 : 65450827] 16.3660338034962 π/3-合同数
(1388264/832195, 197230215/347066, 163893620330899/288826589870)
 [0, -474, 0, -168507, 0]
898704
 1 [-163891265459206734225 : -2341887918990565949640 : 847656828312912773] 26.9472495544465 2π/3-合同数
(3853270092568/269661629195, 808984887585/12193892698, 245047486237088621023939/3288224971171694118110)
 237
238 [0, 476, 0, -169932, 0]
2718912
 1 [336 : 5880 : 1] 3.18125983279919 π/3-合同数
(35/2, 272/5, 481/10)
 [0, -476, 0, -169932, 0]
2718912
 1 [9027454 : 54861120 : 12167] 11.6329357780029 2π/3-合同数
(61920/10189, 1212491/7740, 12600538801/78862860)
 238
239 [0, 478, 0, -171363, 0]
2741808
 1 [-4538131807289520 : -281264871585149569 : 90397942272000] 22.0541954364528 π/3-合同数
(3692546880/239390281, 57214277159/923136720, 12350178816323985121/220989958802218320)
 [0, -478, 0, -171363, 0]
1370904
 1 [12185936772612673 : 357438033037734194 : 8369136358127] 20.8523417599582 2π/3-合同数
(29843685742/1017444239, 2034888478/62434489, 3408223775089802402/63523611147958871)
 239
241 [0, 482, 0, -174243, 0]
1393944
 2 [-1 : 418 : 1],
[-48 : 3060 : 1]
 5.16939037917251,
5.56223664033853
 π/3-合同数
(482/209, 418, 87122/209),
(3856/255, 255/4, 58849/1020)
 [0, -482, 0, -174243, 0]
2787888
 0 - - 2π/3-非合同数
 241
246 [0, 492, 0, -181548, 0]
968256
 1 [-722 : 3344 : 1] 5.35192342475356 π/3-合同数
(88/19, 2337/11, 43927/209)
 [0, -492, 0, -181548, 0]
968256
 0 - - 2π/3-非合同数
 246
247 [0, 494, 0, -183027, 0]
2928432
 0 - - π/3-非合同数
 [0, -494, 0, -183027, 0]
1464216
 0 - - 2π/3-非合同数
 247
249 [0, 498, 0, -186003, 0]
496008
 0 - - π/3-非合同数
 [0, -498, 0, -186003, 0]
992016
 1 [40164367320 : 1408609833147 : 21952000] 12.7772190097778 2π/3-合同数
(850080/29933, 2484439/70840, 116769666187/2120453720)
 249
251 [0, 502, 0, -189003, 0]
3024048
 1 [-169480772245842055614824208 : -2892871293631318325349363340 : 306671853680199307675467] 37.1686951433061 π/3-合同数
(115180162560473935/6747905771378772, 26991623085515088/458885109802685, 162298200121430813224408458546739/3096513480837319613605317602820)
 		[0, -502, 0, -189003, 0]
1512024
 0 - - 2π/3-非合同数
 251
253 [0, 506, 0, -192027, 0]
1536216
 1 [-1683165 : -33338888 : 3375] 8.93242507570844 π/3-合同数
(30008/1515, 34845/682, 46100569/1033230)
 [0, -506, 0, -192027, 0]
3072432
 0 - - 2π/3-非合同数
 253
254 [0, 508, 0, -193548, 0]
3096768
 0 - - π/3-非合同数
 [0, -508, 0, -193548, 0]
3096768
 0 - - 2π/3-非合同数
 254
255 [0, 510, 0, -195075, 0]
1040400
 0 - - π/3-非合同数
 [0, -510, 0, -195075, 0]
520200
 1 [1215 : 28350 : 1] 3.07281225170115 2π/3-合同数
(70/3, 306/7, 1238/21)
 255
257 [0, 514, 0, -198147, 0]
1585176
 1 [-794803017890099270497 : 7662748614342988723170 : 1130691962493104257] 33.5982392682311 π/3-合同数
(277424966334210/28775340491761, 14790525012765154/138712483167105, 407725549272133480274601165794/3991498933591112625438721905)
 		[0, -514, 0, -198147, 0]
3170352
 1 [-193256223291189881989643345 : -2059147627128834381574205056 : 837183471064530448972625] 42.2586534224427 2π/3-合同数
(1437995456709056128/134959517915173985, 34684596104199714145/359498864177264032, 4959742650580832078087652816755348929/48517793400416164272273014562607520)
 		257
258 [0, 516, 0, -199692, 0]
1065024
 1 [-67616400 : 2322137880 : 300763] 13.0040560570309 π/3-合同数
(6683920/222427, 667281/19430, 140069508187/4321756610)
 [0, -516, 0, -199692, 0]
1065024
 2 [-108 : 3780 : 1],
[-2166 : 94240 : 27]
 3.82522909730195,
6.9613413948018
 2π/3-合同数
(1032/35, 35, 1957/35),
(7353/310, 2480/57, 1043521/17670)
 258
259 [0, 518, 0, -201243, 0]
3219888
 2 [-189 : 7056 : 1],
[-588 : 9702 : 1]
 3.1636052577423,
4.18876241222982
 π/3-合同数
(111/4, 112/3, 403/12),
(33/2, 2072/33, 3721/66)
 [0, -518, 0, -201243, 0]
1609944
 1 [-7 : 1176 : 1] 3.47140514488447 2π/3-合同数
(37/6, 168, 1027/6)
 259
262 [0, 524, 0, -205932, 0]
3294912
 1 [-5020134028072050 : -433878770688094160 : 170040671646201] 26.621104885275 π/3-合同数
(1544801761005/127398572587, 1019188580696/11792379855, 121761099292140101478727/1502332360930694034885)
 [0, -524, 0, -205932, 0]
3294912
 1 [-747638523042000400816896 : -20048590210097338580081440 : 5050493888094694057689] 33.9501916312696 2π/3-合同数
(4195605715999565/156459702547896, 1251677620383168/32027524549615, 287618431755327917056964664547/5011016964378199705365860040)
 		262
263 [0, 526, 0, -207507, 0]
3320112
 1 [-405947904643675157696061386400 : -176971286805694390250366680418365 : 370767281462722245658544013312] 45.9412811593841 π/3-合同数
(13319896070364174720/5519727033070218301, 1451688209697467413163/3329974017591043680, 7990837526914682951173967782718395912513/18380547604318726457069188640026387680)
 		[0, -526, 0, -207507, 0]
1660056
 1 [147224334493717651849 : 2553320769369725867210 : 143140204107613727] 26.9453628972106 2π/3-合同数
(9384935865770/541134883513, 1082269767026/17842083395, 684904981096338135414962/9654973719582556566635)
 263
265 [0, 530, 0, -210675, 0]
1685400
 0 - - π/3-非合同数
 [0, -530, 0, -210675, 0]
3370800
 0 - - 2π/3-非合同数
 265
266 [0, 532, 0, -212268, 0]
3396288
 2 [-76 : 4332 : 1],
[2850 : 48735 : 8]
 2.60342585503228,
4.34082159042908
 π/3-非合同数
(56/3, 57, 151/3),
(171/10, 560/9, 5011/90)
 [0, -532, 0, -212268, 0]
3396288
 0 - - 2π/3-非合同数
 266
267 [0, 534, 0, -213867, 0]
1140624
 0 - - π/3-非合同数
 [0, -534, 0, -213867, 0]
570312
 0 - - 2π/3-非合同数
 267
269 [0, 538, 0, -217083, 0]
1736664
 0 - - π/3-非合同数
 [0, -538, 0, -217083, 0]
3473328
 1 [-9598129725725 : -202543561107992 : 51686703125] 21.9990240667427 2π/3-合同数
(3990141272/189084725, 50863791025/997535318, 12108727458137907379/188618691281817550)
 269
271 [0, 542, 0, -220323, 0]
3525168
 0 - - π/3-非合同数
 [0, -542, 0, -220323, 0]
1762584
 0 - - 2π/3-非合同数
 271
273 [0, 546, 0, -223587, 0]
596232
 0 - - π/3-非合同数
 [0, -546, 0, -223587, 0]
1192464
 1 [-48 : 3060 : 1] 5.0878225536663 2π/3-合同数
(1456/85, 255/4, 25099/340)
 273
274 [0, 548, 0, -225228, 0]
3603648
 1 [24178750 : 1189904800 : 12167] 12.6302390399531 π/3-合同数
(2284475/102578, 820624/16675, 73009661953/1710488150)
 [0, -548, 0, -225228, 0]
3603648
 1 [-6704664333703650 : -191501287139229760 : 45030624645033] 25.8170261543846 2π/3-非合同数
(311926908832/10920893805, 1496162451285/38990863604, 24771427115270590845553/425815080784523573220)
 274
277 [0, 554, 0, -230187, 0]
1841496
 1 [59125005975 : 8182942216280 : 3176523] 13.5135824029354 π/3-合同数
(177135/22126, 24515608/177135, 527444361817/3919289010)
 [0, -554, 0, -230187, 0]
3682992
 0 - - 2π/3-非合同数
 277
278 [0, 556, 0, -231852, 0]
3709632
 2 [-17872064 : 616726320 : 68921],
[104623215 : 18582918479 : 3375]
 8.27655569827706,
10.8344232606907
 π/3-合同数
(9184/285, 39615/1148, 10935907/327180),
(316920/50621, 50621/285, 2518539541/14426985)
 [0, -556, 0, -231852, 0]
3709632
 0 - - 2π/3-非合同数
 278
281 [0, 562, 0, -236883, 0]
1895064
 1 [-83640436890793 : 2034987758290086 : 177220220113] 23.0569785404381 π/3-合同数
(16676536818/685425659, 385209220358/8338268409, 228766930958897435042/5715263119157706531)
 [0, -562, 0, -236883, 0]
3790128
 1 [-76207898480713 : -4444444689899200 : 1293960004273] 24.1198365266932 2π/3-合同数
(256070640739/13286512400, 53146049600/911283419, 847256458710464126641/12107778446457895600)
 281
282 [0, 564, 0, -238572, 0]
1272384
 1 [882 : 30240 : 1] 4.05482363057771 π/3-合同数
(329/10, 240/7, 2353/70)
 [0, -564, 0, -238572, 0]
1272384
 0 - - 2π/3-非合同数
 282
283 [0, 566, 0, -240267, 0]
3844272
 0 - - π/3-非合同数
 [0, -566, 0, -240267, 0]
1922136
 1 [-578553027812839215687808078454053767592803347217 : 111666298608188039199808290231481366776455103792960 : 91081195500629389693941293876356815342909503299] 77.0412833192927 2π/3-合同数
(8336213853255693561029619143492073/1421351101863672271844645882030320, 5685404407454689087378583528121280/29456586053907044385263671885131, 8206489829471061654245310124464102283589072244675124688716460791363/41868151044862859486615597765456505550019585430237270712099171920)
 		283
285 [0, 570, 0, -243675, 0]
649800
 1 [-675 : 10800 : 1] 2.84575737910363 π/3-合同数
(16, 285/4, 259/4)
 [0, -570, 0, -243675, 0]
1299600
 1 [-21481875 : -1130652000 : 300763] 7.75531433801976 2π/3-合同数
(5025/232, 17632/335, 5143249/77720)
 285
286 [0, 572, 0, -245388, 0]
3926208
 1 [-351 : 10647 : 1] 3.46269889924748 π/3-合同数
(91/3, 264/7, 727/21)
 [0, -572, 0, -245388, 0]
3926208
 1 [302016 : 31082480 : 27] 5.87584088102499 2π/3-合同数
(1056/95, 1235/12, 124147/1140)
 286
287 [0, 574, 0, -247107, 0]
3953712
 1 [-10474863205476537 : 1498041794734039520 : 843185481401817] 22.7860095126677 π/3-合同数
(4912879419/612025880, 100372244320/701839917, 59781009261018768673/429544192821051960)
 [0, -574, 0, -247107, 0]
1976856
 1 [-189 : 4410 : 1] 2.9452577973126 2π/3-合同数
(70/3, 246/5, 962/15)
 287
290 [0, 580, 0, -252300, 0]
4036800
 0 - - π/3-非合同数
 [0, -580, 0, -252300, 0]
4036800
 0 - - 2π/3-非合同数
 290
291 [0, 582, 0, -254043, 0]
1354896
 0 - - π/3-非合同数
 [0, -582, 0, -254043, 0]
677448
 0 - - 2π/3-非合同数
 291
293 [0, 586, 0, -257547, 0]
2060376
 0 - - π/3-非合同数
 [0, -586, 0, -257547, 0]
4120752
 1 [-3919491111651923106816969956465 : -168965844906960622719058399420792 : 38739898896341937640666204625] 49.4740335157241 2π/3-合同数
(33742366850562226798955/1241129709179816962082, 4964518836719267848328/115161661606014425935, 8775424190816014653519699823490934939010579/142930559577737177248817414166498892396670)
 		293
295 [0, 590, 0, -261075, 0]
4177200
 0 - - π/3-非合同数
 [0, -590, 0, -261075, 0]
2088600
 0 - - 2π/3-非合同数
 295
298 [0, 596, 0, -266412, 0]
4262592
 1 [4460460529073741312684 : 39882615215103002232940 : 13981479351701200039] 30.121357639277 π/3-合同数
(37966115905235/4246119782371, 33968958258968/254806147015, 139650796656670297341121447/1081937421510124831272565)
 		[0, -596, 0, -266412, 0]
4262592
 1 [1452480991249456 : 34935712712325480 : 1060895910599] 20.0822605720386 2π/3-合同数
(792014985/157656142, 1261249136/25449765, 260848191039220387/4012311764706630)
 298
299 [0, 598, 0, -268203, 0]
4291248
 1 [-529 : 12696 : 1] 2.90746179009681 π/3-合同数
(24, 299/6, 259/6)
 [0, -598, 0, -268203, 0]
2145624
 2 [1573 : 44616 : 1],
[24375 : 67600 : 27]
 4.18841961125426,
5.8876162643918
 2π/3-合同数
(312/11, 253/6, 4057/66),
(208/75, 1725/4, 129793/300)
 299
301 [0, 602, 0, -271803, 0]
2174424
 1 [-147 : 7056 : 1] 3.24281986946994 π/3-合同数
(301/12, 48, 499/12)
 [0, -602, 0, -271803, 0]
4348848
 0 - - 2π/3-非合同数
 301
302 [0, 604, 0, -273612, 0]
4377792
 0 - - π/3-非合同数
 [0, -604, 0, -273612, 0]
4377792
 2 [-294 : 1680 : 1],
[1344169454215 : 29097445508979 : 1045678375]
 4.46453483404075,
16.0913734400367
 2π/3-合同数
(40/7, 1057/5, 7501/35),
(5295219/244615, 295494920/5295219, 89653314400861/129528999568)
 302
303 [0, 606, 0, -275427, 0]
1468944
 0 - - π/3-非合同数
 [0, -606, 0, -275427, 0]
734472
 1 [21720007198725 : 246065149651410 : 21554582687] 16.3288448840313 2π/3-合同数
(15921894/1405415, 283893830/2653649, 421704652489694/3729478109335)
 303
305 [0, 610, 0, -279075, 0]
2232600
 1 [-675 : 12600 : 1] 3.98571145515518 π/3-合同数
(56/3, 915/14, 2449/42)
 [0, -610, 0, -279075, 0]
4465200
 1 [-300 : 1350 : 1] 4.25258880047338 2π/3-合同数
(9/2, 2440/9, 4921/18)
 305
307 [0, 614, 0, -282747, 0]
4523952
 0 - - π/3-非合同数
 [0, -614, 0, -282747, 0]
2261976
 1 [-190868035852437659984523972647126588969085569475942747558484251 : 3828992718819348628664735350409317371054657084754537262664168960 : 839606663916255131233629453203134869561700032594164838684793] 97.1299345590183 2π/3-合同数
(1554187066030024990763933888897630293291520/77473281989390317606942303155473391372537, 23784297570742827505331287068730331151368859/388546766507506247690983472224407573322880, 2207428826951499770334385168994110639601993097123550107259753336398450469559988134083/30101993207701828856631222340137670868648101079833427463929017975293415629065746560)
 		307
309 [0, 618, 0, -286443, 0]
763848
 1 [-17115813 : -398486088 : 29791] 11.3132597639648 π/3-合同数
(103208/4433, 1369797/25802, 5271999379/114380266)
 [0, -618, 0, -286443, 0]
1527696
 1 [-10164 : -965250 : 343] 9.09108500081663 2π/3-合同数
(63448/4875, 14625/154, 76650883/750750)
 309
310 [0, 620, 0, -288300, 0]
4612800
 1 [320 : 2000 : 1] 4.14505303272626 π/3-合同数
(25/4, 992/5, 3907/20)
 [0, -620, 0, -288300, 0]
4612800
 1 [-270 : 3600 : 1] 2.86776429836149 2π/3-合同数
(40/3, 93, 301/3)
 310
311 [0, 622, 0, -290163, 0]
4642608
 1 [-4653686802119148001122203520 : 609068158523863193714893422803 : 265027343345025306329088000] 41.1335548647109 π/3-合同数
(392157163901099520/41258032807267237, 12831248203060110707/98039290975274880, 511254020634042785132699172265782241/4044908283459109750885347693106560)
 		[0, -622, 0, -290163, 0]
2321304
 1 [15800977 : 336395638 : 12167] 6.94022658697803 2π/3-合同数
(23014/1081, 2162/37, 2859602/39997)
 311
313 [0, 626, 0, -293907, 0]
2351256
 0 - - π/3-非合同数
 [0, -626, 0, -293907, 0]
4702512
 0 - - 2π/3-非合同数
 313
314 [0, 628, 0, -295788, 0]
4732608
 0 - - π/3-非合同数
 [0, -626, 0, -293907, 0]
4702512
 0 - - 2π/3-非合同数
 314
317 [0, 634, 0, -301467, 0]
2411736
 0 - - π/3-非合同数
 [0, -634, 0, -301467, 0]
4823472
 1 [-631633822975419852906565 : -27353977442588180664978712 : 5498526330079068242125] 39.0433534125107 2π/3-合同数
(1058440223751014455/36149491205453738, 144597964821814952/3338928150634115, 7634499859772267051321719378651699/120700553816989853946670997071870)
 		317
318 [0, 636, 0, -303372, 0]
1617984
 1 [405069930 : -8747386128 : 857375] 12.764966485446 π/3-合同数
(576472/26695, 4244505/72059, 99283868173/1923615005)
 [0, -636, 0, -303372, 0]
1617984
 0 - - 2π/3-非合同数
 318
319 [0, 638, 0, -305283, 0]
4884528
 0 - - π/3-非合同数
 [0, -638, 0, -305283, 0]
2442264
 2 [-121 : 5082 : 1],
[-44109 : -2908178 : 729]
 3.10844400984591,
6.50500330613894
 2π/3-合同数
(638/21, 42, 1322/21),
(2574/133, 7714/117, 1205102/15561)
 319
321 [0, 642, 0, -309123, 0]
824328
 0 - - π/3-非合同数
 [0, -642, 0, -309123, 0]
1648656
 1 [87535388354016240 : 5786978340352478841 : 17253512704000] 22.5602008994729 2π/3-合同数
(10618069440/546699569, 58496853883/884839120, 37569507951929042827/483741165538339280)
 321
322 [0, 644, 0, -311052, 0]
4976832
 1 [1922 : 94240 : 1] 6.35401153905012 π/3-合同数
(4991/190, 1520/31, 250321/5890),
 [0, -644, 0, -311052, 0]
4976832
 1 [-76664 : -2746695 : 512] 8.6075907341345 2π/3-合同数
(10605/296, 54464/1515, 27875569/448440)
 322
323 [0, 646, 0, -312987, 0]
5007792
 1 [22550185683532 : 1005467646233175 : 14598344384] 20.3287937342336 π/3-合同数
(4327277968/149337975, 2538745575/56937868, 333217446399022849/8502985907937300)
 [0, -646, 0, -312987, 0]
2503896
 0 - - 2π/3-非合同数
 323
326 [0, 652, 0, -318828, 0]
5101248
 0 - - π/3-非合同数
 [0, -652, 0, -318828, 0]
5101248
 0 - - 2π/3-非合同数
 326
327 [0, 654, 0, -320787, 0]
1710864
 2 [-18900 : 1514430 : 343],
[28215740 : 810260438 : 42875]
 8.64932077403692,
9.33848778241196
 π/3-合同数
(91560/5609, 5609/70, 28796281/392630),
(86437/3010, 36120/793, 95217691/2386930)
 [0, -654, 0, -320787, 0]
855432
 1 [-38159219475 : -2245121073720 : 498677257] 17.3864052533469 2π/3-合同数
(346180185/15571654, 62286616/1058655, 1196028725391889/16485009365370)
 327
329 [0, 658, 0, -324723, 0]
2597784
 1 [336 : 1764 : 1] 4.00585103572447 π/3-合同数
(21/4, 752/3, 2977/12)
 [0, -658, 0, -324723, 0]
5195568
 1 [28749 : 290080 : 27] 6.66300469701205 2π/3-合同数
(1120/111, 5217/40, 602737/4440)
 329
330 [0, 660, 0, -326700, 0]
1742400
 1 [-396 : 13068 : 1] 1.97478017756162 π/3-合同数
(33, 40, 37)
 [0, -660, 0, -326700, 0]
1742400
 0 - - 2π/3-非合同数
 330
331 [0, 662, 0, -328683, 0]
5258928
 0 - - π/3-非合同数
 [0, -662, 0, -328683, 0]
2629464
 1 [-652197365884338406153821316788727808 : -23763281628664626243987478596033284160 : 4252184017503945155454959044279793] 52.2416579388087 2π/3-合同数
(71466775848620975029504/1966724732594472168045, 650985886488770287622895/17866693962155243757376, 2214587886746832419369430030700588514681392579/35138868905067042216598925560544762680249920)
 		331
334 [0, 668, 0, -334668, 0]
5354688
 1 [726 : 22176 : 1] 5.05795303879341 π/3-合同数
(336/11, 1837/42, 17953/462)
 [0, -668, 0, -334668, 0]
5354688
 1 [1293402122451602 : 47698482217034880 : 592915705201] 24.4435918219088 2π/3-合同数
(389254597117/10744798680, 85958389440/2330865851, 1585631925900999322417/25044684319081876680)
 334
335 [0, 670, 0, -336675, 0]
5386800
 1 [-2997998508800 : 140848481835600 : 15419656169] 16.3913782751225 π/3-合同数
(42213440/1480017, 99161139/2110672, 128066336684323/3123830441424)
 [0, -670, 0, -336675, 0]
2693400
 1 [-245 : 5250 : 1] 3.99569569288571 2π/3-合同数
(150/7, 938/15, 7934/105)
 335
337 [0, 674, 0, -340707, 0]
2725656
 0 - - π/3-非合同数
 [0, -674, 0, -340707, 0]
5451312
 0 - - 2π/3-非合同数
 337
339 [0, 678, 0, -344763, 0]
1838736
 0 - - π/3-非合同数
 [0, -678, 0, -344763, 0]
919368
 0 - - 2π/3-非合同数
 339
341 [0, 682, 0, -348843, 0]
2790744
 0 - - π/3-非合同数
 [0, -682, 0, -348843, 0]
5581488
 1 [-132 : 35090 : 27] 6.18281750914485 2π/3-合同数
(744/145, 1595/6, 233539/870)
 341
345 [0, 690, 0, -357075, 0]
952200
 0 - - π/3-非合同数
 [0, -690, 0, -357075, 0]
1904400
 1 [-245 : 5600 : 1] 4.19549683608839 2π/3-合同数
(160/7, 483/8, 4171/56)
 345
346 [0, 692, 0, -359148, 0]
5746368
 1 [21789004 : 1040930620 : 12167] 8.11755623071666 π/3-合同数
(6808/235, 40655/851, 8336167/199985)
 [0, -692, 0, -359148, 0]
5746368
 1 [12439397974166096 : 17665482663170760 : 11966561852617] 21.3191463814335 2π/3-合同数
(910614045/641221682, 5129773456/5263665, 3291721164674263267/3375176124784530)
 346
347 [0, 694, 0, -361227, 0]
5779632
 1 [-56160879324591488869078800 : 1304068752047854309819442660 : 82179038746664305614531] 36.2491098322862 π/3-合同数
(61595026221261809/2652644524435620, 10610578097742480/177507280176547, 24576578353906687584366164944723/470863714807776874169135404140)
 		[0, -694, 0, -361227, 0]
2889816
 0 - - 2π/3-非合同数
 347
349 [0, 698, 0, -365403, 0]
2923224
 1 [154654829482921 : 7642478301828120 : 80007275629] 17.946684592632 π/3-合同数
(43697569/1546830, 2159374680/43697569, 2902573794037561/67592710656270)
 [0, -698, 0, -365403, 0]
5846448
 0 - - 2π/3-非合同数
 349
353 [0, 706, 0, -373827, 0]
2990616
 1 [-346304186915711360222153 : 11109824797409340113460750 : 735549656676038400617] 37.9765298424126 π/3-合同数
(56721408304649250/1768062191965087, 1248251907527351422/28360704152324625, 1976938539102276240632894247803714/50143488749232421109007890367375)
 		[0, -706, 0, -373827, 0]
5981232
 1 [-50636289679856793337705636381487420762409124465 : 4148312900234038875578106962591991551757926004416 : 1011508745470156569794878404277689858155556625] 74.8302059742014 2π/3-合同数
(2516714300712666019700445329036815/146018825721912966497120270609488, 584075302887651865988481082437952/7129502268307835749859618495855, 95529724044950207923821893895260115838946941269386184103696856449/1041041549200025046649553011867155571938413960196870638651672240)
 		353
354 [0, 708, 0, -375948, 0]
2005056
 1 [-294 : 12096 : 1] 4.20703039469898 π/3-合同数
(413/12, 288/7, 3211/84)
 [0, -708, 0, -375948, 0]
2005056
 0 - - 2π/3-非合同数
 354
355 [0, 710, 0, -378075, 0]
6049200
 0 - - π/3-非合同数
 [0, -710, 0, -378075, 0]
3024600
 1 [57245 : 13610400 : 1] 8.65549296691186 2π/3-合同数
(7597/1272, 25440/107, 32773681/136104)
 355
357 [0, 714, 0, -382347, 0]
1019592
 1 [-6820961 : 316639960 : 29791] 8.04346241453824 π/3-合同数
(3999/130, 61880/1333, 7088467/173290)
 [0, -714, 0, -382347, 0]
2039184
 1 [78920919 : 3869351640 : 24389] 7.7476017845124 2π/3-合同数
(6699/230, 15640/319, 5019331/73370)
 357
358 [0, 716, 0, -384492, 0]
6151872
 1 [-36631650 : 314842960 : 35937] 12.7070435692453 π/3-合同数
(211304/24585, 4400715/26413, 105511758727/649363605)
 [0, -716, 0, -384492, 0]
6151872
 1 [-20697217120669305959671943199636262420689137573670 : 162988292597706542131587790979791820856373869231744 : 60343634882116776734759104801072678318196382625] 76.443551574948 2π/3-合同数
(916002625462280261552232510403008/116319429574555311777902379369065, 20821177893845400808244525907062635/114500328182785032694029063800376, 2476015434871438862542098096375299577046149458511614701814632413667/13318612860320934390447989448254630299225420513613107925989768440)
 		358
359 [0, 718, 0, -386643, 0]
6186288
 1 [-418746285518454192154569296093293004640 : 6518889639890560127504829130324012595179 : 463440685687259783086247007522816000] 55.9175277593999 π/3-合同数
(14790514459872748901484011/950080969783479283025760, 3800323879133917132103040/41199204623600971870429, 3347791017464392186324132888672647208941865704481/39142580283098814923729392595951842857789251040)
 		[0, -718, 0, -386643, 0]
3093144
 1 [1057071171 : 45552259662 : 389017] 15.1333453323045 2π/3-合同数
(115153558/3455641, 6911282/160381, 36777124532882/554219159221)
 359
362 [0, 724, 0, -393132, 0]
6290112
 0 - - π/3-非合同数
 [0, -724, 0, -393132, 0]
6290112
 0 - - 2π/3-非合同数
 362
365 [0, 730, 0, -399675, 0]
3197400
 0 - - π/3-非合同数
 [0, -730, 0, -399675, 0]
6394800
 1 [-65533 : -7367528 : 2197] 10.8989554455379 2π/3-合同数
(336895/25942, 103768/923, 2860132339/23944466)
 365
367 [0, 734, 0, -404067, 0]
6465072
0 - - π/3-非合同数
 [0, -734, 0, -404067, 0]
3232536
 0 - - 2π/3-非合同数
 367
370 [0, 740, 0, -410700, 0]
6571200
 1 [-999 : 12321 : 1] 2.9505235641054 π/3-合同数
(37/3, 120, 343/3)
 [0, -740, 0, -410700, 0]
6571200
 1 [357795173190 : 6670375120800 : 258474853] 16.7191081284779 2π/3-合同数
(51385680/2756299, 101983063/1284642, 319261995500083/3540857459958)
 370
371 [0, 742, 0, -412923, 0]
6606768
 1 [568575 : 17912440 : 729] 8.83080162142211 π/3-合同数
(26936/855, 45315/962, 34185043/822510)
 [0, -742, 0, -412923, 0]
3303384
 0 - - 2π/3-非合同数
 371
373 [0, 746, 0, -417387, 0]
3339096
 1 [4505800651595964548215921714972844975 : -264792866299791169773233010325825633560 : 1578862060837247651821567199785187] 52.4470691131015 π/3-合同数
(37504979781498752123135/1477251432485340408078, 2204059137268127888852376/37504979781498752123135, 2828409769287277600022265452118958914910071097/55404285107552760872223962709210109804684530)
 		[0, -746, 0, -417387, 0]
6678192
 0 - - 2π/3-非合同数
 373
374 [0, 748, 0, -419628, 0]
6714048
 2 [-594 : 17424 : 1],
[-50562 : 4147568 : 729]
 2.59019942383108,
9.55413173211532
 π/3-合同数
(88/3, 51, 133/3),
(89199/4891, 39128/477, 174046327/2333007)
 [0, -748, 0, -419628, 0]
6714048
 0 - - 2π/3-非合同数
 374
377 [0, 754, 0, -426387, 0]
3411096
 1 [974454 : 70071625 : 216] 9.52887099904533 π/3-合同数
(280488/13375, 13375/186, 159344857/2487750)
 [0, -754, 0, -426387, 0]
6822192
 1 [-169 : 6760 : 1] 3.29179511005632 2π/3-合同数
(377/10, 40, 673/10)
 377
379 [0, 758, 0, -430923, 0]
6894768
 0 - - π/3-非合同数
 [0, -758, 0, -430923, 0]
3447384
 1 [-429975 : -15190560 : 2197] 10.452420375284 2π/3-合同数
(48224/1365, 517335/12056, 1116795619/16456440)
 379
381 [0, 762, 0, -435483, 0]
1161288
 1 [-17199 : -4083912 : 2197] 9.28941291836338 π/3-合同数
(34671/5402, 21608/91, 115181299/491582)
 [0, -762, 0, -435483, 0]
2322576
 1 [-173196983590742967345 : -401077067886223399944 : 456186241330335125] 26.5580242892797 2π/3-合同数
(456629842136/197186320555, 591558961665/898877642, 116853102876854452264963/177246374855134531310)
 381
382 [0, 764, 0, -437772, 0]
7004352
 1 [-203054633401016028985020 : 9155224591592703969294721 : 719462835026627112000] 33.5075088596244 π/3-合同数
(780814403557920/23039898453697, 4400620604656127/97601800444740, 91437896902485004235336858881/2248735571144808292897203780)
 		[0, -764, 0, -437772, 0]
7004352
 1 [1119304108293098605362587269493541907200 : 3038021922954001983635836253610581582880 : 972013174012990724064234170799418559] 55.940458588067 2π/3-合同数
(1887795325903934889865633/695523968387365727282520, 5564191747098925818260160/9883745161800706229663, 3879391621664043119698341871297970074600995966721/6874381657465053354933547943662772484005390760)
 		382
383 [0, 766, 0, -440067, 0]
7041072
 1 [-63897112799422770753544211924668576686271440646545758531482891833792460685610000 : 7530705194293275817958464966578900958052112958217757343204762380412420129094025575 : 1910473254890278840960578881987356335475067227635046859519738590144912792154112] 120.750499790697 π/3-合同数
(18199441005311520618236955798275171404200213915902400/1400082718853431016636560859648316304372131401575919, 536231681320864079371802809245305144574526326803576977/4549860251327880154559238949568792851050053478975600, 712981747408362253396440740268203234227734532315778725207128576557759534804723272343824366985817986078113/6370180711082293415818590422820448617180225812038467831152771755247045132668953030422409822373748576400)
 		[0, -766, 0, -440067, 0]
3520536
 1 [351517904308562231329 : 50289266361427500579170 : 16539079737866687] 28.0547849389156 2π/3-合同数
(967140471454/90314741185, 69181091747710/483570235727, 6494554198029792977996642/43673520684453445316495)
 383
385 [0, 770, 0, -444675, 0]
3557400
 2 [-825 : 18150 : 1],
[-605 : 18150 : 1]
 1.6351512594455,
2.07545424571126
 π/3-合同数
(22, 70, 62),
(30, 154/3, 134/3)
 [0, -770, 0, -444675, 0]
7114800
 2 [-189 : 7056 : 1],
[1215 : 10800 : 1]
 3.38867263853499,
4.38802440340413
 2π/3-合同数
(112/3, 165/4, 817/12),
(80/9, 693/4, 6403/36)
 385
386 [0, 772, 0, -446988, 0]
7151808
 0 - - π/3-非合同数
 [0, -772, 0, -446988, 0]
7151808
 0 - - 2π/3-非合同数
 386
389 [0, 778, 0, -453963, 0]
3631704
 0 - - π/3-非合同数
 [0, -778, 0, -453963, 0]
7263408
 1 [-1090937188350289465 : -5771143584325980952 : 2855129341389625] 29.6865400538247 2π/3-合同数
(2081132050088/393402852335, 153033709558315/520283012522, 60752523883292579324178739/204680821147601321938870)
 389
390 [0, 780, 0, -456300, 0]
2433600
 1 [-6318:150579:8] 4.22341384083391 π/3-合同数
(143/6, 720/11, 3787/66)
 [0, -780, 0, -456300, 0]
2433600
 2 [-351 : 4563 : 1],
[-300 : 6300 : 1]
 2.43335248769159,
3.22753787598749
 2π/3-合同数
(13, 120, 127),
(21, 520/7, 607/7)
 390
391 [0, 782, 0, -458643, 0]
7338288
 0 - - π/3-非合同数
 [0, -782, 0, -458643, 0]
3669144
 0 - - 2π/3-非合同数
 391
393 [0, 786, 0, -463347, 0]
1235592
 0 - - π/3-非合同数
 [0, -786, 0, -463347, 0]
2471184
 1 [117692709576537358007936249511600 : 2836709028174198017680123418884035 : 71411923175468368867113807872] 45.2823748293019 2π/3-合同数
(283194143873471555153/11749490623692507280, 140993887484310087360/2161787357812759963, 2033057742642205135763732583678240704011/25399900291038022458115628100270030640)
 		393
394 [0, 788, 0, -465708, 0]
7451328
 1 [-6 : 1680 : 1] 4.59634903300891 π/3-合同数
(197/35, 280, 9703/35)
 [0, -788, 0, -465708, 0]
7451328
 1 [100658227047153104 : 2976759065443647000 : 52789237764697] 22.7344578620049 2π/3-合同数
(64736145375/2189033602, 17512268816/328609875, 52322105140003427107/719338058324019750)
 394
395 [0, 790, 0, -468075, 0]
7489200
 1 [20311820 : 569199750 : 29791] 12.2635290192831 π/3-合同数
(314475/11222, 3546152/62895, 34463613619/705807690)
 [0, -790, 0, -468075, 0]
3744600
 0 - - 2π/3-非合同数
 395
397 [0, 794, 0, -472827, 0]
3782616
 1 [1491055524077438387753104215353859049375 : 18207060773203443180241973865560408704200 : 3411826618544679301373829945392546347] 56.134967587158 π/3-合同数
(29035468589312040291471432/2377841013085902186201775, 2377841013085902186201775/18284300119214131165914, 5408259569175621714004375211228875844524614226777/43477158719038811768881685956679172604006297350)
 		[0, -794, 0, -472827, 0]
7565232
 0 - - 2π/3-非合同数
 397
398 [0, 796, 0, -475212, 0]
7603392
 0 - - π/3-非合同数
 [0, -796, 0, -475212, 0]
7603392
 2 [-6 : 1680 : 1],
[113032490225906 : 3196495762009760 : 61023377953]
 4.60257545323292,
22.790415481211
 2π/3-合同数
(199/35, 280, 9901/35),
(13339519760/471703781, 93869052419/1667439970, 58653195191109140161/786537738439526570)
 398
399 [0, 798, 0, -477603, 0]
2547216
 2 [-684 : 19494 : 1],
[-108 : 7722 : 1]
 2.42146630204849,
4.99332241065957
 π/3-合同数
(57/2, 56, 97/2),
(3192/143, 143/2, 18121/286)
 [0, -798, 0, -477603, 0]
1273608
 1 [-147 : 7056 : 1] 2.85727082394679 2π/3-合同数
(133/4, 48, 283/4)
 399
401 [0, 802, 0, -482403, 0]
3859224
 1 [-2761785644777769608969 : 126226196151984884126862 : 9104309573265848969] 35.0336018783154 π/3-合同数
(60903724886720642/1735397701979679, 3470795403959358/75939806591921, 5460072831694890746019784429682/131785765848400982355787573359)
 		[0, -802, 0, -482403, 0]
7718448
 1 [-9156507660781 : -1793803867350880 : 743923693981] 24.0785032769335 2π/3-合同数
(115504091729/14107111480, 56428445920/288040129, 813187880049707133841/4063414210516580920)
 401
402 [0, 804, 0, -484812, 0]
2585664
 1 [-5160000 : 467521200 : 79507] 12.3399206163587 π/3-合同数
(2304800/129867, 389601/4300, 46440993067/558428100)
 [0, -804, 0, -484812, 0]
2585664
 0 - - 2π/3-非合同数
 402
403 [0, 806, 0, -487227, 0]
7795632
 0 - - π/3-非合同数
 [0, -806, 0, -487227, 0]
3897816
 1 [-96227107586024932290800 : -6427341258323382576581220 : 1058260886270694039149] 32.8585326261917 2π/3-合同数
(197312670713680/8175679629201, 3294798890568003/49328167678420, 32901800072998093131352368403/403291295634269578609542420)
 		403
406 [0, 812, 0, -494508, 0]
7912128
 1 [-42 : 4704 : 1] 2.71562602211006 π/3-合同数
(29/2, 112, 211/2)
 [0, -812, 0, -494508, 0]
7912128
 1 [-294 : 7056 : 1] 2.78129539511806 2π/3-合同数
(24, 203/3, 247/3)
 406
407 [0, 814, 0, -496947, 0]
7951152
 1 [507 : 9360 : 1] 5.58716533835951 π/3-合同数
(240/13, 5291/60, 62833/780)
 [0, -814, 0, -496947, 0]
3975576
 1 [-11955625 : -695725800 : 103823] 7.60040489597045 2π/3-合同数
(1175/42, 68376/1175, 3757417/49350)
 407
409 [0, 818, 0, -501843, 0]
4014744
 2 [507 : 9282 : 1],
[-169 : 10166 : 1]
 5.41583865801735,
5.84748913453782
 π/3-合同数
(238/13, 10634/119, 126482/1547),
(10634/391, 782/13, 265202/5083)
 [0, -818, 0, -501843, 0]
8029488
 2 [-561600 : -16750440 : 2197],
[2381092301082960 : 23083370178661133 : 1833767424000]
 11.2370703244746,
20.1168321589009
 2π/3-合同数
(46529/1560, 2552160/46529, 5399792641/72585240),
(2587856293/266942160, 1067768640/6327277, 293562196791495361/1689016989298320)
 409
410 [0, 820, 0, -504300, 0]
8068800
 0 - - π/3-非合同数
 [0, -820, 0, -504300, 0]
8068800
 2 [-50 : 4800 : 1],
[1280 : 10400 : 1]
 3.58903738185223,
4.9907942369475
 2π/3-合同数
(205/12, 96, 1267/12),
(65/8, 2624/13, 21427/104)
 410
411 [0, 822, 0, -506763, 0]
2702736
 0 - - π/3-非合同数
 [0, -822, 0, -506763, 0]
1351368
 0 - - 2π/3-非合同数
 411
413 [0, 826, 0, -511707, 0]
4093656
 0 - - π/3-非合同数
 [0, -826, 0, -511707, 0]
8187312
 1 [52593443 : 1738886520 : 24389] 11.8581329498304 2π/3-合同数
(488040/14761, 870899/17430, 18627275011/257284230)
 413
415 [0, 830, 0, -516675, 0]
8266800
 0 - - π/3-非合同数
 [0, -830, 0, -516675, 0]
4133400
 0 - - 2π/3-非合同数
 415
417 [0, 834, 0, -521667, 0]
1391112
 0 - - π/3-非合同数
 [0, -834, 0, -521667, 0]
2782224
 1 [-1259193547843899100401800090025 : -2121316467782716670516441052480 : 3024788887603673400809809673] 41.634899648346 2π/3-合同数
(1173507261153902528/696582897476622715, 2089748692429868145/2110624570420688, 1456923194823009982844905658187614539/1470224978748994968872753006727920)
 		417
418 [0, 836, 0, -524172, 0]
8386752
 1 [-550 : 19360 : 1] 3.19235074087163 π/3-合同数
(176/5, 95/2, 427/10)
 [0, -836, 0, -524172, 0]
8386752
 1 [891683456180 : 19064122404657 : 551368000] 15.5788565525037 2π/3-合同数
(39989433/1870420, 14963360/191337, 32496264287521/357880551540)
 418
419 [0, 838, 0, -526683, 0]
8426928
 1 [-22943690090152257726418313555869241070077888269968 : -813911598703930699044828190042507086430100129812540 : 42071666421101730288089013675805554287039837403] 72.5083342770928 π/3-合同数
(489557315286474387855437008160635/13800333280894405628045826501492, 55201333123577622512183306005968/1168394547223089231158560878665, 686854495770850887279012657626560590878901172274241640464472819/16124234155258348559981486865102614844567138049614714453468180)
 		[0, -838, 0, -526683, 0]
4213464
 0 - - 2π/3-非合同数
 419
421 [0, 842, 0, -531723, 0]
4253784
 1 [338529747220829896557132713144865 : 944106393401297941253558238539288 : 800399141204084260719467824125] 45.905870599078 π/3-合同数
(240971625874849053976/86405583274268391105, 86405583274268391105/143094789711905614, 7448743787897755246202742120118517861689/12364188768565984368435621215397163470)
 		[0, -842, 0, -531723, 0]
8507568
 0 - - 2π/3-非合同数
 421
422 [0, 844, 0, -534252, 0]
8548032
 0 - - π/3-非合同数
 [0, -844, 0, -534252, 0]
8548032
 0 - - 2π/3-非合同数
 421
422 [0, 844, 0, -534252, 0]
8548032
 0 - - π/3-非合同数
 [0, -844, 0, -534252, 0]
8548032
 0 - - 2π/3-非合同数
 422
426 [0, 852, 0, -544428, 0]
2903616
 1 [450 : 4320 : 1] 3.74441890015039 π/3-合同数
(48/5, 355/2, 1729/10)
 [0, -852, 0, -544428, 0]
2903616
 0 - - 2π/3-非合同数
 426
427 [0, 854, 0, -546987, 0]
8751792
 0 - - π/3-非合同数
 [0, -854, 0, -546987, 0]
4375896
 1 [-529661799239079945200 : -18165304700454806747580 : 2199135161314518461] 28.7906047381561 2π/3-合同数
(43558349057769/1270069173860, 5080276695440/102010185147, 9489262608811209966076243/129559991574955932657420)
 427
429 [0, 858, 0, -552123, 0]
1472328
 1 [441 : 3024 : 1] 4.429098688965 π/3-合同数
(48/7, 1001/4, 6913/28)
 [0, -858, 0, -552123, 0]
2944656
 1 [1875 : 50400 : 1] 5.84657309755567 2π/3-合同数
(672/25, 3575/56, 112993/1400)
 429
430 [0, 860, 0, -554700, 0]
8875200
 1 [-150 : 47600 : 27] 5.81992658617414 π/3-合同数
(645/119, 952/3, 112333/357)
 [0, -860, 0, -554700, 0]
8875200
 1 [-2820800 : -317288400 : 68921] 7.99156727219558 2π/3-合同数
(6560/429, 18447/164, 8502883/70356)
 430
431 [0, 862, 0, -557283, 0]
8916528
 1 [-631700631798526385486775602970054597360 : -28167885096403142441672585417318299235047 : 1698342896295907511648355787516416000] 56.4464076024436 π/3-合同数
(5289499434643754312093760/136810756004616841009223, 58965435837989858474975113/1322374858660938578023440, 7587954003799217494555320547541752387041704160001/180915104134901349005983101767570153258150187120)
 		[0, -862, 0, -557283, 0]
4458264
 1 [15963909 : 80343354 : 12167] 12.1765634708471 2π/3-合同数
(55678/11063, 9536306/27839, 106283640482/307982857)
 431
433 [0, 866, 0, -562467, 0]
4499736
 0 - - π/3-非合同数
 [0, -866, 0, -562467, 0]
8999472
 2 [-6481728 : -251170920 : 29791],
[8294038767405600 : 109798872615401905 : 5785468305408]
 3.0148570341632,
20.919692521972
 2π/3-合同数
(317135/8184, 14174688/317135, 187722626017/2595432840),
(7762959529/586402080, 2345608320/17928313, 1450075929211003873/10513200034091040)
 433
434 [0, 868, 0, -565068, 0]
9041088
 0 - - π/3-非合同数
 [0, -868, 0, -565068, 0]
9041088
 2 [-42 : 4704 : 1],
[-432 : 1224 : 1]
 2.75620702958323,
5.52314139963612
 2π/3-合同数
(31/2, 112, 241/2),
(17/6, 10416/17, 62641/102)
 434
435 [0, 870, 0, -567675, 0]
3027600
 0 - - π/3-非合同数
 [0, -870, 0, -567675, 0]
1513800
 0 - - 2π/3-非合同数
 435
437 [0, 874, 0, -572907, 0]
4583256
 0 - - π/3-非合同数
 [0, -874, 0, -572907, 0]
9166512
 1 [-6162623056969060 : -194422582078684918 : 22991097212875] 24.2873862816451 2π/3-合同数
(95810635361/3036914870, 279396168040/5042665019, 1167616241459544869491/15314144380629932530)
 437
438 [0, 876, 0, -575532, 0]
3069504
 1 [486 : 6480 : 1] 3.80050984900572 π/3-合同数
(40/3, 657/5, 1879/15)
 [0, -876, 0, -575532, 0]
3069504
 0 - - 2π/3-非合同数
 438
439 [0, 878, 0, -578163, 0]
9250608
 0 - - π/3-非合同数
 [0, -878, 0, -578163, 0]
4625304
 0 - - 2π/3-非合同数
 439
442 [0, 884, 0, -586092, 0]
9377472
 1 [109330 : 3607136 : 125] 6.47658726129787 π/3-合同数
(4784/145, 2465/46, 312289/6670)
 [0, -884, 0, -586092, 0]
9377472
 1 [38997774512 : 352853207595 : 28094464] 5.9745530875209 2π/3-合同数
(7555905/835088, 86849152/444465, 74262826017649/371167387920)
 442
443 [0, 886, 0, -588747, 0]
9419952
 1 [-28568790582282261728836074147623944471566801189770849370000 : -1903918255786767597455822252353674498618501038901380322885100 : 180213846435771575965825365995169373810632279538083380259] 87.037858191456 π/3-合同数
(76342916777415691705161650862079275600/2871186965431728107420731777547617229, 1271935825686255551587384177453594432447/19085729194353922926290412715519818900, 3184139319078717216705149562443160184253946095148166937155001213392059409363/54798696888588780846402715795994996133316442512718598245054513820599828100)
 		[0, -886, 0, -588747, 0]
4709976
 0 - - 2π/3-非合同数
 443
445 [0, 890, 0, -594075, 0]
4752600
 1 [1032295 : 30962000 : 1331] 9.62375772020435 π/3-合同数
(45200/1507, 134123/2260, 175047361/3405820)
 [0, -890, 0, -594075, 0]
9505200
 2 [-25205 : -1085448 : 125],
[10580 : 1038450 : 1]
 7.67724871500415,
9.24215113450927
 2π/3-合同数
(16376/903, 4515/46, 4501219/41538),
(157975/3822, 15288/355, 99177139/1356810)
 445
446 [0, 892, 0, -596748, 0]
9547968
 0 - - π/3-非合同数
 [0, -892, 0, -596748, 0]
9547968
 0 - - 2π/3-非合同数
 446
447 [0, 894, 0, -599427, 0]
3196944
 0 - - π/3-非合同数
 [0, -894, 0, -599427, 0]
1598472
 1 [-122747319888917 : -2000384584132990 : 316796140513] 18.5622062976149 2π/3-合同数
(1221268070/74939281, 449635686/4098215, 36456474192796334/307117285483415)
 447
449 [0, 898, 0, -604803, 0]
4838424
 1 [-36044577577 : 1660334051262 : 95443993] 18.3154183218431 π/3-合同数
(3644638066/93476751, 186953502/4058617, 16300528729084562/379386330713367)
 [0, -898, 0, -604803, 0]
9676848
 1 [-1856261 : -154997920 : 24389] 12.7244573728085 2π/3-合同数
(3294313/153160, 612640/7337, 107965705681/1123734920)
 449
451 [0, 902, 0, -610203, 0]
9763248
 0 - - π/3-非合同数
 [0, -902, 0, -610203, 0]
4881624
 1 [17157164904 : 90578462535 : 12487168] 15.3136034851009 2π/3-合同数
(1833535/347304, 56957856/166685, 19936259636899/57890367240)
 451
453 [0, 906, 0, -615627, 0]
1641672
 1 [-2097783684675 : -80450826202440 : 3803721481] 19.3996873129582 π/3-合同数
(422347304/11012855, 4988823315/105586826, 50567996642943379/1162812404648230)
 [0, -906, 0, -615627, 0]
3283344
 1 [-18278538549433969190673520846755225 : -1009673554642448078057686829005373640 : 121945111127027048858634432027413] 48.6120524694655 2π/3-合同数
(1414541518772942525505/43121813620709216666, 26045575426908366866264/471513839590980841835, 1566967017069986719504098999247084101340099/20332531910427258367887143029016092022110)
 		453
454 [0, 908, 0, -618348, 0]
9893568
 1 [-101572306726 : -24702951710064 : 9568634867] 20.6119136190901 π/3-合同数
(2357075611/315668469, 2525347752/10383593, 785221329854295607/3277772905029117)
 [0, -908, 0, -618348, 0]
9893568
 1 [-60500880081600 : -5258587953591280 : 835098514713] 18.8749994058666 2π/3-合同数
(200393760/9591263, 2177216701/25049220, 23792624600596963/240253656964860)
 454
455 [0, 910, 0, -621075, 0]
9937200
 1 [-270725 : 29811600 : 4913] 5.57086185352193 π/3-合同数
(595/36, 1872/17, 62947/612)
 [0, -910, 0, -621075, 0]
4968600
 1 [-325 : 8450 : 1] 1.73458104795459 2π/3-合同数
(26, 70, 86)
 455
457 [0, 914, 0, -626547, 0]
5012376
 2 [-121 : 9350 : 1],
[15770430200 : 4749693037515 : 175616]
 5.90845982756747,
13.3376758355706
 π/3-合同数
(10054/425, 850/11, 320594/4675),
(175840/28971, 13239747/43960, 379762754113/1273565160)
 [0, -914, 0, -626547, 0]
10024752
 0 - - 2π/3-非合同数
 457
458 [0, 916, 0, -629292, 0]
10068672
 0 - - π/3-非合同数
 [0, -916, 0, -629292, 0]
10068672
 0 - - 2π/3-非合同数
 458
461 [0, 922, 0, -637563, 0]
5100504
 0 - - π/3-非合同数
 [0, -922, 0, -637563, 0]
10201008
 1 [-663847569087608940867520577811114901180935582 : -26671077749261260641783655003261064687207695095 : 2880659376812513569202034729969725387719496] 65.2535031252141 2π/3-合同数
(332029961786814341623544414305/8264281071376410348634663058, 33057124285505641394538652232/720238528821723083782092005, 444020941294661004362820145625132710768179343407922027219/5952253640617359250637991648819403671952400457530651290)
 		461
462 [0, 924, 0, -640332, 0]
3415104
 1 [-63 : 6615 : 1] 3.14513076232411 π/3-合同数
(88/5, 105, 487/5)
 [0, -924, 0, -640332, 0]
3415104
 0 - - 2π/3-非合同数
 462
463 [0, 926, 0, -643107, 0]
10289712
 0 - - π/3-非合同数
 [0, -926, 0, -643107, 0]
5144856
 0 - - 2π/3-非合同数
 463
465 [0, 930, 0, -648675, 0]
1729800
 0 - - π/3-非合同数
 [0, -930, 0, -648675, 0]
3459600
 1 [-8097975 : -591591600 : 79507] 6.86353227221714 2π/3-合同数
(1935/76, 9424/129, 868369/9804)
 465
466 [0, 932, 0, -651468, 0]
10423488
 1 [7420641573758 : 137789694003520 : 13127554279] 21.09478479511 π/3-合同数
(1737180640/93555581, 21798450373/217147580, 1879365756188945137/20315368009643980)
 [0, -932, 0, -651468, 0]
10423488
 1 [-12654633750 : -450238947200 : 47832147] 17.1814243104647 2π/3-合同数
(53920832/1515525, 353117325/6740104, 782924426377153/10214796114600)
 466
467 [0, 934, 0, -654267, 0]
10468272
 1 [-187793825928014840239135512450240629856769825261278374805570489958800 : 3100204907084629452393824179601012653340653206538804764450238463490020 : 155896531867230138970666216470984841916374486172831348556453367899] 101.252961069531 π/3-合同数
(768004998125236210923726398308208951631148301/46521633650791093446775665275173267084469940, 186086534603164373787102661100693068337879760/1644550317184659980564724621644987048460703, 8099733501709749185195280867747376652913791091290578849529191160607356811106758526428403/76507167376357043994465865217981511802790733259617331982890264396988722910953674767820)
 		[0, -934, 0, -654267, 0]
5234136
 0 - - 2π/3-非合同数
 467
469 [0, 938, 0, -659883, 0]
5279064
 1 [-1323 : 14112 : 1] 4.17803993562256 π/3-合同数
(32/3, 1407/8, 4099/24)
 [0, -938, 0, -659883, 0]
10558128
 0 - - 2π/3-非合同数
 469
470 [0, 940, 0, -662700, 0]
10603200
 0 - - π/3-非合同数
 [0, -940, 0, -662700, 0]
10603200
 0 - - 2π/3-非合同数
 470
471 [0, 942, 0, -665523, 0]
3549456
 0 - - π/3-非合同数
 [0, -940, 0, -662700, 0]
10603200
 0 - - 2π/3-非合同数
 471
473 [0, 946, 0, -671187, 0]
5369496
 1 [9785266491 : 54780045650 : 20346417] 11.0550219783459 π/3-合同数
(126850/22659, 498498/1475, 11203086182/33422025)
 [0, -946, 0, -671187, 0]
10738992
 1 [2039051221720205 : 20121962234002112 : 1365783682375] 22.9833651115849 2π/3-合同数
(14152187456/1434106415, 61666575845/321640624, 90798005982341143819/461266882203002960)
 473
474 [0, 948, 0, -674028, 03594816]
3594816
 1 [-11648277900 : 232773761340 : 10218313] 15.7297953201454 π/3-合同数
(3057181/152985, 290059560/3057181, 40518286508461/467702835285)
 [0, -948, 0, -674028, 0]
3594816
 0 - - 2π/3-非合同数
 474
478 [0, 956, 0, -685452, 0]
10967232
 1 [-525155138286375420420 : -20136989676869813014993 : 861225399603288000] 29.1565962283387 π/3-合同数
(55441877085529/1445875828260, 11567006626080/231974381111, 15166422191011888324507681/335406150423968023996860)
 [0, -956, 0, -685452, 0]
10967232
 1 [9139281794513733111714382430539639044115200 : 523406928476990656195263773642447801501222880 : 2080927500416836360032615315493285563839] 62.0748273930102 2π/3-合同数
(1613612585654198872194281280/48332447020945187156329511, 11551454838005899730362753129/201701573206774859024285160, 774178187757256814192223718549837573266209342334247681/9748730601057743115265731605094625815885356187356760)
 		478
479 [0, 958, 0, -688323, 0]
11013168
 1 [-110370425193495712649480815200 : 15482728374264638455629088413089 : 3007639736791256996352000000] 42.7609353405239 π/3-合同数
(2306868797159203200/168897090815827439, 80901706500781343281/576717199289800800, 13049819499439492505292626083043192641/97405857183499137523522805484151200)
 		[0, -958, 0, -688323, 0]
5506584
 1 [708464279959951 : 13969903755687038 : 406704318929] 18.3663206446731 2π/3-合同数
(2064174818/104681761, 209363522/2154671, 24445705602170882/225554754655631)
 479
481 [0, 962, 0, -694083, 0]
5552664
 2 [-1369 : 13690 : 1],
[-1200 : 22140 : 1]
 2.91813399997087,
6.79118341657227
 π/3-合同数
(10, 962/5, 938/5),
(369/20, 38480/369, 711361/7380)
 [0, -962, 0, -694083, 0]
11105328
 2 [-432 : 6300 : 1],
[-57245 : -559824 : 125]
 6.30638763373619,
9.38851210439296
 2π/3-合同数
(175/12, 23088/175, 293569/2100),
(5232/535, 257335/1308, 141220369/699780)
 481
482 [0, 964, 0, -696972, 0]
11151552
 0 - - π/3-非合同数
 [0, -964, 0, -696972, 0]
11151552
 0 - - 2π/3-非合同数
 482
483 [0, 966, 0, -699867, 0]
3732624
 0 - - π/3-非合同数
 [0, -966, 0, -699867, 0]
1866312
 0 - - 2π/3-非合同数
 483
485 [0, 970, 0, -705675, 0]
5645400
 0 - - π/3-非合同数
 [0, -970, 0, -705675, 0]
11290800
 1 [-620177 : -41575352 : 4913] 11.7472713472431 2π/3-合同数
(1574795/54418, 217672/3247, 15067386259/176695246)
 485
487 [0, 974, 0, -711507, 0]
11384112
 0 - - π/3-非合同数
 [0, -970, 0, -705675, 0]
11290800
 1 [-620177 : -41575352 : 4913] 11.7472713472431 2π/3-合同数
(1574795/54418, 217672/3247, 15067386259/176695246)
 487
489 [0, 978, 0, -717363, 0]
1912968
 0 - - π/3-非合同数
 [0, -978, 0, -717363, 0]
3825936
 1 [-3514128085197780561 : -30526082325136712160 : 7466212702103849] 23.8119323181534 2π/3-合同数
(108518384480/12492513733, 37477541199/166439240, 477475836714078203467/2079244491410082920)
 489
491 [0, 982, 0, -723243, 0]
11571888
 1 [-812530368 : -51022382840 : 3581577] 10.7766931694151 π/3-合同数
(63648/2035, 999185/15912, 1760929699/32380920)
 [0, -982, 0, -723243, 0]
5785944
 0 - - 2π/3-非合同数
 491
493 [0, 986, 0, -729147, 0]
5833176
 1 [7038573105141227 : 102506659489661960 : 12784998722887] 20.1825951841813 π/3-合同数
(2040854120/140134319, 2382283423/17593570, 317413514952361513/2465462950728830)
 [0, -986, 0, -729147, 0]
11666352
 0 - - 2π/3-非合同数
 493
494 [0, 988, 0, -732108, 0]
11713728
 0 - - π/3-非合同数
 [0, -988, 0, -732108, 0]
11713728
 0 - - 2π/3-非合同数
 494
497 [0, 994, 0, -741027, 0]
5928216
 1 [-432113535 : 24890681214 : 1520875] 13.0863095910076 π/3-合同数
(6907590/200147, 2802058/48645, 488860405874/9736150815)
 [0, -994, 0, -741027, 0]
11856432
 1 [-13266264368 : -885294997895 : 99897344] 17.1333376927519 2π/3-合同数
(266319296/8939755, 62578285/937744, 717669513739249/8383201612720)
 497
498 [0, 996, 0, -744012, 0]
3968064
 1 [2045654818409070 : 86857330437387279 : 1509003523000] 20.1824127639099 π/3-合同数
(4344612839/102323870, 2455772880/52344733, 240241669854984187/5356115654676710)
 [0, -996, 0, -744012, 0]
3968064
 2 [-450 : 6480 : 1],
[2500870510 : 134526077296 : 614125]
 3.85693776436593,
11.4485631995678
 2π/3-合同数
(72/5, 415/3, 2191/15),
(107355/2899, 1924936/35785, 8205947239/103740715)
 498
499 [0, 998, 0, -747003, 0]
11952048
 0 - - π/3-非合同数
 [0, -998, 0, -747003, 0]
5976024
 1 [-407338070075014457683507264767836735577870765163275 : 1691220153052301680575647776191991686939620350634880 : 823373833671904262693467087592671828814287493257] 79.2153476134005 2π/3-合同数
(4683762451422079564668551899554176/1128105500758591499538512728523655, 562924644878537158269717851533303845/1170940612855519891167137974888544, 637798277577450931781594486176318083937242997701853985170116622412419/1320944546423948289961663697763411222378060794306208961628292508320)
 		499


[2007.12.19追記]
"heegner.gp"により、E2π/3,358の有理点群の生成元を求めることができた。よって、358は(2π/3)-合同数である。

[2007.12.22追記]
"heegner.gp"により、E2π/3,499の有理点群の生成元を求めることができた。よって、499は(2π/3)-合同数である。

[2008.09.19追記]
"heegner.gp"により、E2π/3,353の2-isogenyであるE~2π/3,353: y^2=x^3+2*706x^2+(706^2+4*373827)xの有理点群の生成元
    [2432186632670539844993372526770517712222638560385391341205812446907177287237213947373363363409920 : 223186210025312368295735146383022089762984589127919403759820091880927960724303257393153773974209792 : 362391192890183490879305841837383641166349291797653027465936226273946679407802676995672151375]
    height 149.6604119484029270832043855
を求めることができた。
これにより、E2π/3,353の有理点群の生成元
    [-50636289679856793337705636381487420762409124465 : 4148312900234038875578106962591991551757926004416 : 1011508745470156569794878404277689858155556625]
    height 74.83020597420146354160219276
を求めることができた。よって、353は(2π/3)-合同数である。

[関連リンク]

[参考文献]

Last Update: 2014.02.22
H.Nakao

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