実二次体のイデアル類群
実二次体のイデアル類群
ここでは平方因子を持たない正整数$d$に対して実二次体$K=\mathbb{Q}(\sqrt{d})$のイデアル類群の数表を記す. 具体的には
- $d \leq 10000$ の場合
- $d \leq 100000$ が素数の場合
のそれぞれの場合において$K$のイデアル類群の数表を記述する. なお, イデアル類群の計算には計算機代数ソフトウェアである Magma を用いて計算した.
記号
- $d > 0$を$1$以外の平方因子を持たない整数,
- $K = \mathbb{Q}(\sqrt{d})$を実二次体,
- $\operatorname{Cl}(K)$を$K$のイデアル類群,
- $\# \operatorname{Cl}(K)$を$K$の類数
とする. また - $C_n$を$n$次巡回群
とする.
1. 実二次体のイデアル類群
$d \leq 10000$ に対する実二次体 $K=\mathbb{Q}(\sqrt{d})$ のイデアル類群は以下のように分類される.
$d \equiv 1 \pmod{4}$の場合
類数 1 (1038)
$\operatorname{Cl}(K) = 1$ (1038)
5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, 201, 209, 213, 217, 233, 237, 241, 249, 253, 269, 277, 281, 293, 301, 309, 313, 317, 329, 337, 341, 349, 353, 373, 381, 389, 393, 397, 409, 413, 417, 421, 433, 437, 449, 453, 457, 461, 489, 497, 501, 509, 517, 521, 537, 541, 553, 557, 569, 573, 581, 589, 593, 597, 601, 613, 617, 633, 641, 649, 653, 661, 669, 673, 677, 681, 701, 709, 713, 717, 721, 737, 749, 753, 757, 769, 773, 781, 789, 797, 809, 813, 821, 829, 849, 853, 857, 869, 877, 881, 889, 893, 913, 917, 921, 929, 933, 937, 941, 953, 973, 977, 989, 997, 1013, 1021, 1033, 1041, 1049, 1057, 1061, 1069, 1077, 1081, 1097, 1109, 1117, 1121, 1133, 1137, 1141, 1149, 1153, 1169, 1177, 1181, 1193, 1201, 1213, 1217, 1237, 1249, 1253, 1273, 1277, 1289, 1293, 1301, 1317, 1321, 1329, 1333, 1337, 1349, 1357, 1361, 1381, 1389, 1397, 1401, 1409, 1433, 1437, 1441, 1453, 1457, 1461, 1473, 1477, 1481, 1493, 1497, 1501, 1529, 1541, 1549, 1553, 1561, 1569, 1577, 1589, 1597, 1609, 1613, 1621, 1633, 1637, 1657, 1661, 1669, 1673, 1689, 1693, 1697, 1709, 1713, 1721, 1733, 1741, 1753, 1757, 1777, 1789, 1793, 1797, 1801, 1817, 1821, 1829, 1837, 1841, 1857, 1861, 1873, 1877, 1889, 1893, 1909, 1913, 1933, 1941, 1949, 1969, 1973, 1977, 1981, 1993, 1997, 2017, 2033, 2049, 2053, 2069, 2073, 2077, 2113, 2129, 2137, 2141, 2149, 2157, 2161, 2181, 2189, 2201, 2217, 2221, 2229, 2237, 2253, 2269, 2273, 2281, 2293, 2297, 2309, 2317, 2321, 2333, 2341, 2357, 2361, 2369, 2377, 2381, 2389, 2393, 2413, 2417, 2433, 2437, 2441, 2449, 2453, 2461, 2469, 2473, 2477, 2481, 2489, 2497, 2513, 2517, 2521, 2537, 2549, 2569, 2573, 2577, 2593, 2609, 2617, 2621, 2629, 2633, 2641, 2649, 2653, 2657, 2661, 2681, 2689, 2693, 2721, 2729, 2733, 2741, 2749, 2753, 2757, 2761, 2773, 2789, 2797, 2801, 2833, 2837, 2841, 2861, 2869, 2881, 2893, 2897, 2901, 2909, 2921, 2933, 2949, 2953, 2957, 2969, 2973, 3001, 3013, 3017, 3037, 3041, 3049, 3053, 3057, 3061, 3073, 3089, 3093, 3097, 3101, 3109, 3113, 3117, 3149, 3153, 3169, 3189, 3193, 3197, 3209, 3217, 3241, 3257, 3269, 3273, 3301, 3309, 3313, 3317, 3329, 3337, 3353, 3361, 3369, 3373, 3377, 3389, 3397, 3401, 3409, 3413, 3421, 3433, 3437, 3449, 3453, 3457, 3461, 3469, 3473, 3489, 3493, 3513, 3517, 3521, 3529, 3533, 3541, 3557, 3561, 3581, 3593, 3613, 3617, 3629, 3637, 3641, 3661, 3669, 3673, 3677, 3693, 3697, 3701, 3709, 3713, 3733, 3749, 3761, 3769, 3777, 3781, 3793, 3797, 3817, 3821, 3829, 3833, 3837, 3841, 3849, 3853, 3881, 3901, 3909, 3917, 3921, 3929, 3937, 3949, 3953, 3989, 4013, 4021, 4037, 4049, 4057, 4061, 4073, 4093, 4101, 4109, 4117, 4129, 4133, 4153, 4157, 4169, 4177, 4189, 4197, 4201, 4213, 4217, 4237, 4241, 4249, 4253, 4261, 4269, 4273, 4289, 4297, 4309, 4313, 4317, 4333, 4337, 4341, 4349, 4373, 4377, 4393, 4397, 4413, 4417, 4421, 4429, 4449, 4457, 4461, 4497, 4501, 4513, 4517, 4529, 4533, 4541, 4549, 4561, 4569, 4577, 4593, 4601, 4609, 4613, 4621, 4629, 4637, 4657, 4661, 4673, 4677, 4681, 4701, 4713, 4721, 4733, 4737, 4741, 4769, 4781, 4789, 4793, 4801, 4813, 4817, 4821, 4829, 4837, 4861, 4873, 4877, 4881, 4897, 4909, 4937, 4957, 4969, 4973, 4989, 4993, 4997, 5001, 5009, 5021, 5029, 5033, 5053, 5077, 5093, 5097, 5101, 5113, 5129, 5137, 5149, 5153, 5169, 5173, 5177, 5189, 5197, 5201, 5209, 5221, 5233, 5237, 5257, 5277, 5293, 5309, 5349, 5357, 5361, 5377, 5381, 5393, 5401, 5413, 5433, 5437, 5441, 5449, 5461, 5469, 5489, 5493, 5501, 5509, 5533, 5541, 5549, 5557, 5561, 5569, 5573, 5581, 5601, 5609, 5633, 5641, 5653, 5657, 5669, 5677, 5689, 5693, 5701, 5717, 5721, 5737, 5749, 5753, 5761, 5773, 5789, 5793, 5801, 5813, 5833, 5849, 5857, 5861, 5869, 5873, 5881, 5893, 5897, 5909, 5921, 5937, 5953, 5961, 5969, 5977, 5981, 5997, 6009, 6013, 6017, 6029, 6033, 6037, 6041, 6049, 6073, 6077, 6089, 6101, 6117, 6121, 6169, 6173, 6181, 6189, 6193, 6197, 6217, 6221, 6229, 6233, 6249, 6257, 6261, 6269, 6277, 6281, 6297, 6301, 6313, 6317, 6329, 6333, 6337, 6349, 6353, 6361, 6373, 6377, 6389, 6393, 6397, 6421, 6429, 6433, 6449, 6457, 6469, 6473, 6493, 6509, 6521, 6529, 6533, 6537, 6541, 6553, 6569, 6577, 6581, 6589, 6593, 6609, 6621, 6629, 6653, 6661, 6673, 6677, 6689, 6701, 6709, 6717, 6729, 6733, 6737, 6753, 6761, 6769, 6781, 6793, 6797, 6801, 6821, 6829, 6833, 6841, 6857, 6861, 6869, 6881, 6913, 6917, 6933, 6937, 6941, 6961, 6973, 6977, 7001, 7009, 7013, 7017, 7037, 7041, 7061, 7069, 7073, 7097, 7109, 7113, 7121, 7129, 7133, 7149, 7153, 7169, 7177, 7181, 7193, 7197, 7201, 7213, 7217, 7233, 7237, 7253, 7269, 7277, 7297, 7309, 7313, 7321, 7333, 7341, 7349, 7357, 7369, 7377, 7393, 7401, 7409, 7417, 7433, 7457, 7477, 7489, 7493, 7509, 7517, 7529, 7541, 7549, 7561, 7577, 7589, 7593, 7597, 7609, 7613, 7617, 7621, 7629, 7637, 7649, 7653, 7661, 7669, 7681, 7697, 7717, 7729, 7737, 7741, 7757, 7773, 7781, 7789, 7793, 7829, 7841, 7849, 7853, 7877, 7901, 7909, 7933, 7937, 7941, 7949, 7961, 7977, 7981, 7989, 7993, 7997, 8009, 8013, 8053, 8061, 8081, 8089, 8093, 8117, 8129, 8133, 8137, 8141, 8153, 8157, 8161, 8189, 8193, 8197, 8201, 8209, 8213, 8221, 8233, 8237, 8257, 8261, 8269, 8273, 8293, 8297, 8301, 8317, 8329, 8341, 8353, 8369, 8377, 8389, 8401, 8409, 8413, 8417, 8429, 8453, 8457, 8461, 8509, 8513, 8521, 8529, 8537, 8549, 8553, 8557, 8561, 8573, 8609, 8617, 8629, 8641, 8657, 8669, 8677, 8681, 8693, 8709, 8717, 8737, 8741, 8753, 8777, 8781, 8797, 8809, 8813, 8817, 8821, 8849, 8861, 8873, 8881, 8889, 8893, 8913, 8921, 8929, 8933, 8941, 8953, 8969, 8977, 8981, 8997, 9001, 9013, 9017, 9033, 9037, 9041, 9053, 9057, 9069, 9097, 9109, 9121, 9137, 9157, 9161, 9173, 9201, 9209, 9221, 9233, 9237, 9241, 9249, 9253, 9257, 9277, 9313, 9329, 9337, 9341, 9349, 9353, 9357, 9377, 9397, 9421, 9433, 9437, 9449, 9461, 9473, 9481, 9489, 9493, 9497, 9501, 9521, 9533, 9557, 9561, 9569, 9573, 9589, 9601, 9609, 9613, 9617, 9629, 9637, 9641, 9649, 9661, 9677, 9689, 9697, 9713, 9721, 9733, 9753, 9761, 9769, 9777, 9781, 9793, 9817, 9853, 9857, 9901, 9913, 9917, 9921, 9929, 9941, 9949, 9957, 9961, 9969, 9973, 9977, 9989, 9993
類数 2 (543)
$\operatorname{Cl}(K) \cong C_2$ (543)
65, 85, 105, 165, 185, 205, 221, 265, 273, 285, 305, 345, 357, 365, 377, 385, 429, 465, 481, 485, 493, 533, 545, 561, 565, 609, 629, 645, 665, 685, 705, 741, 745, 805, 861, 865, 885, 949, 957, 965, 969, 1001, 1005, 1037, 1065, 1073, 1085, 1113, 1157, 1165, 1173, 1185, 1189, 1205, 1209, 1241, 1245, 1261, 1265, 1281, 1285, 1309, 1353, 1385, 1405, 1417, 1465, 1469, 1505, 1513, 1517, 1533, 1537, 1545, 1565, 1581, 1585, 1605, 1645, 1649, 1653, 1685, 1717, 1729, 1749, 1769, 1781, 1833, 1853, 1865, 1869, 1905, 1921, 1945, 1961, 1965, 1985, 2001, 2013, 2037, 2041, 2045, 2065, 2085, 2093, 2105, 2109, 2117, 2121, 2165, 2173, 2185, 2193, 2245, 2257, 2261, 2265, 2285, 2301, 2329, 2337, 2345, 2353, 2365, 2373, 2397, 2409, 2445, 2485, 2509, 2553, 2561, 2581, 2585, 2613, 2685, 2697, 2701, 2717, 2737, 2765, 2769, 2785, 2813, 2821, 2829, 2845, 2865, 2877, 2885, 2905, 2929, 2937, 2965, 2977, 2985, 3005, 3009, 3065, 3077, 3085, 3133, 3157, 3165, 3205, 3233, 3237, 3265, 3277, 3289, 3293, 3297, 3333, 3345, 3349, 3365, 3385, 3405, 3417, 3441, 3497, 3505, 3553, 3565, 3589, 3597, 3605, 3621, 3633, 3649, 3653, 3657, 3665, 3685, 3689, 3729, 3737, 3741, 3745, 3765, 3785, 3801, 3809, 3813, 3857, 3865, 3869, 3893, 3913, 3945, 3961, 3977, 3985, 4029, 4033, 4069, 4085, 4089, 4105, 4121, 4137, 4141, 4145, 4181, 4209, 4233, 4245, 4265, 4277, 4285, 4301, 4369, 4381, 4385, 4405, 4433, 4445, 4453, 4465, 4469, 4521, 4537, 4553, 4565, 4573, 4585, 4589, 4605, 4633, 4645, 4665, 4685, 4697, 4709, 4717, 4773, 4809, 4849, 4885, 4893, 4917, 4921, 4929, 4945, 4965, 4985, 5017, 5057, 5061, 5065, 5069, 5117, 5133, 5141, 5161, 5165, 5205, 5213, 5253, 5285, 5289, 5317, 5321, 5345, 5385, 5389, 5397, 5405, 5429, 5453, 5457, 5465, 5473, 5485, 5505, 5585, 5593, 5597, 5617, 5645, 5649, 5665, 5673, 5681, 5705, 5709, 5729, 5745, 5757, 5765, 5781, 5797, 5829, 5837, 5845, 5885, 5889, 5905, 5917, 5941, 5957, 5965, 5989, 5993, 6005, 6061, 6065, 6109, 6141, 6149, 6161, 6177, 6213, 6265, 6285, 6293, 6341, 6357, 6365, 6369, 6385, 6437, 6441, 6461, 6465, 6477, 6485, 6501, 6505, 6513, 6573, 6585, 6605, 6613, 6617, 6641, 6645, 6657, 6665, 6693, 6697, 6721, 6749, 6773, 6785, 6789, 6817, 6837, 6853, 6865, 6873, 6905, 6965, 6969, 6985, 7005, 7021, 7033, 7081, 7089, 7141, 7145, 7157, 7165, 7185, 7189, 7241, 7257, 7261, 7265, 7285, 7305, 7329, 7337, 7361, 7365, 7373, 7385, 7405, 7413, 7421, 7429, 7445, 7469, 7485, 7501, 7521, 7545, 7553, 7633, 7657, 7689, 7701, 7733, 7765, 7769, 7777, 7797, 7801, 7805, 7809, 7813, 7845, 7869, 7885, 7913, 7945, 7953, 7969, 7973, 7985, 8021, 8029, 8033, 8041, 8045, 8065, 8165, 8169, 8205, 8241, 8249, 8313, 8333, 8337, 8357, 8365, 8393, 8421, 8437, 8445, 8449, 8481, 8485, 8489, 8493, 8497, 8517, 8533, 8565, 8589, 8593, 8621, 8633, 8665, 8697, 8729, 8733, 8749, 8765, 8785, 8805, 8853, 8857, 8877, 8885, 8897, 8917, 8961, 8965, 8985, 8989, 9005, 9077, 9085, 9093, 9105, 9113, 9129, 9141, 9169, 9185, 9193, 9197, 9205, 9269, 9273, 9285, 9309, 9361, 9365, 9373, 9381, 9385, 9389, 9393, 9401, 9417, 9429, 9445, 9453, 9465, 9469, 9485, 9509, 9529, 9541, 9545, 9577, 9581, 9593, 9645, 9665, 9681, 9701, 9705, 9709, 9717, 9737, 9741, 9773, 9785, 9809, 9821, 9861, 9865, 9877, 9885, 9889, 9893, 9965, 9985, 9997
類数 3 (112)
$\operatorname{Cl}(K) \cong C_3$ (112)
229, 257, 321, 469, 473, 733, 761, 993, 1101, 1229, 1257, 1373, 1489, 1509, 1901, 1929, 1957, 2021, 2089, 2101, 2177, 2213, 2429, 2557, 2589, 2677, 2713, 2777, 2857, 2917, 2981, 3173, 3221, 3229, 3261, 3569, 3873, 3877, 3889, 3941, 3957, 3981, 4001, 4193, 4281, 4481, 4493, 4597, 4649, 4729, 4749, 4841, 4853, 4857, 4933, 5081, 5089, 5261, 5281, 5297, 5333, 5477, 5497, 5613, 5637, 5741, 5821, 5853, 6053, 6133, 6209, 6289, 6557, 6637, 6901, 6997, 7053, 7117, 7249, 7273, 7441, 7481, 7537, 7601, 7673, 7721, 7753, 8017, 8057, 8069, 8173, 8373, 8581, 8597, 8637, 8713, 8837, 8909, 9073, 9133, 9149, 9281, 9293, 9301, 9413, 9517, 9749, 9813, 9833, 9869, 9897, 9937
類数 4 (159)
$\operatorname{Cl}(K) \cong C_2 \times C_2$ (56)
1105, 1365, 1885, 2145, 2405, 2465, 2665, 2805, 3045, 3145, 3445, 3485, 3705, 3885, 3965, 4389, 4485, 4505, 4745, 4785, 4845, 5005, 5313, 5365, 5565, 5785, 5945, 6045, 6105, 6205, 6305, 6405, 6545, 6765, 7085, 7161, 7293, 7345, 7565, 7685, 7905, 7917, 8177, 8245, 8265, 8385, 8585, 8645, 8845, 9061, 9165, 9177, 9265, 9605, 9685, 9933
$\operatorname{Cl}(K) \cong C_4$ (103)
145, 445, 505, 689, 777, 793, 897, 901, 905, 1045, 1145, 1221, 1313, 1677, 1745, 2005, 2249, 2289, 2501, 2533, 2545, 2669, 2849, 2945, 3029, 3161, 3245, 3341, 3477, 3545, 3805, 3845, 3905, 4017, 4045, 4053, 4081, 4173, 4705, 4777, 4953, 4981, 5037, 5045, 5105, 5109, 5181, 5217, 5245, 5421, 5545, 5605, 5809, 5817, 5933, 5973, 6001, 6145, 6245, 6445, 6497, 6649, 6745, 6805, 6893, 6981, 6989, 7045, 7077, 7093, 7205, 7289, 7397, 7437, 7505, 7557, 7645, 7761, 7837, 7897, 7957, 8077, 8149, 8305, 8345, 8601, 8605, 8701, 8705, 8801, 8841, 8945, 8949, 9089, 9145, 9213, 9597, 9673, 9789, 9797, 9841, 9845, 9881
類数 5 (37)
$\operatorname{Cl}(K) \cong C_5$ (37)
401, 817, 1093, 1393, 1429, 1641, 1897, 2081, 2153, 3121, 3181, 3253, 3997, 4353, 4357, 4441, 4757, 4889, 5241, 5269, 6113, 6157, 6481, 6949, 7229, 7513, 7817, 7861, 8049, 8121, 8309, 8501, 8689, 9181, 9229, 9757, 9829
類数 6 (40)
$\operatorname{Cl}(K) \cong C_6$ (40)
697, 785, 985, 1345, 1765, 1937, 2233, 2505, 2941, 2993, 3021, 3281, 3305, 3973, 4065, 4765, 5073, 5353, 5369, 5529, 5621, 5685, 5901, 6153, 6185, 6549, 6601, 6681, 6685, 7453, 7473, 7709, 8113, 8277, 8285, 8769, 8789, 9565, 9745, 9905
類数 7 (16)
$\operatorname{Cl}(K) \cong C_7$ (16)
577, 1009, 1601, 1761, 2029, 2913, 4229, 5273, 5417, 8097, 8661, 8773, 9029, 9049, 9101, 9289
類数 8 (30)
$\operatorname{Cl}(K) \cong C_2 \times C_4$ (7)
1785, 4305, 5865, 6409, 6565, 7585, 9345
$\operatorname{Cl}(K) \cong C_8$ (23)
1705, 2605, 2705, 3081, 3201, 4161, 4669, 5249, 5305, 5513, 5713, 6757, 6953, 7449, 7833, 8005, 8105, 8229, 8473, 8653, 9021, 9305, 9953
類数 9 (7)
$\operatorname{Cl}(K) \cong C_9$ (7)
1129, 3137, 4409, 5521, 6809, 7573, 7873
類数 10 (16)
$\operatorname{Cl}(K) \cong C_{10}$ (16)
3129, 3585, 4097, 4321, 4865, 5777, 6085, 6945, 7049, 7221, 7705, 8321, 8569, 9321, 9553, 9669
類数 11 (3)
$\operatorname{Cl}(K) \cong C_{11}$ (3)
1297, 4009, 6081
類数 12 (11)
$\operatorname{Cl}(K) \cong C_2 \times C_{6}$ (5)
4641, 7665, 8745, 8905, 9805
$\operatorname{Cl}(K) \cong C_{12}$ (6)
4345, 5629, 6401, 7745, 7881, 8545
類数 13 (2)
$\operatorname{Cl}(K) \cong C_{13}$ (2)
8101, 8441
類数 14 (4)
$\operatorname{Cl}(K) \cong C_{14}$ (4)
6097, 8185, 8465, 9505
類数 16 (1)
$\operatorname{Cl}(K) \cong C_{16}$ (1)
2305
類数 18 (2)
$\operatorname{Cl}(K) \cong C_{18}$ (2)
7465, 9217
類数 20 (2)
$\operatorname{Cl}(K) \cong C_2 \times C_{10}$ (1)
5185
$\operatorname{Cl}(K) \cong C_{20}$ (1)
3601
類数 21 (1)
$\operatorname{Cl}(K) \cong C_{20}$ (1)
7057
類数 27 (1)
$\operatorname{Cl}(K) \cong C_{27}$ (1)
8761
$d \equiv 2, 3 \pmod{4}$の場合
類数 1 (789)
$\operatorname{Cl}(K) = 1$ (789)
2, 3, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 227, 239, 251, 262, 263, 271, 278, 283, 302, 307, 311, 331, 334, 347, 358, 367, 379, 382, 383, 398, 419, 422, 431, 446, 454, 463, 467, 478, 479, 487, 491, 502, 503, 523, 526, 542, 547, 563, 566, 571, 587, 599, 607, 614, 619, 622, 631, 643, 647, 662, 683, 691, 694, 718, 719, 734, 739, 743, 751, 758, 766, 787, 811, 823, 827, 838, 859, 862, 863, 878, 883, 886, 887, 907, 911, 919, 926, 947, 958, 967, 971, 974, 983, 991, 998, 1006, 1019, 1031, 1039, 1046, 1051, 1063, 1094, 1103, 1123, 1142, 1151, 1163, 1174, 1187, 1198, 1214, 1231, 1238, 1259, 1262, 1279, 1283, 1286, 1291, 1303, 1307, 1318, 1319, 1366, 1382, 1399, 1423, 1427, 1438, 1439, 1447, 1451, 1454, 1459, 1471, 1483, 1487, 1499, 1502, 1511, 1531, 1543, 1559, 1571, 1574, 1579, 1583, 1607, 1619, 1622, 1663, 1667, 1678, 1699, 1718, 1723, 1726, 1747, 1759, 1774, 1783, 1814, 1822, 1823, 1831, 1838, 1867, 1871, 1879, 1894, 1931, 1942, 1951, 1966, 1979, 1982, 1999, 2003, 2011, 2038, 2039, 2062, 2063, 2078, 2083, 2087, 2102, 2111, 2126, 2131, 2174, 2179, 2182, 2203, 2206, 2239, 2243, 2246, 2267, 2287, 2311, 2326, 2339, 2342, 2347, 2351, 2371, 2374, 2383, 2411, 2423, 2446, 2447, 2462, 2503, 2518, 2531, 2539, 2551, 2558, 2566, 2579, 2582, 2591, 2606, 2614, 2647, 2654, 2663, 2671, 2683, 2687, 2699, 2707, 2719, 2731, 2734, 2767, 2791, 2798, 2803, 2819, 2843, 2846, 2851, 2854, 2879, 2887, 2903, 2927, 2939, 2942, 2963, 2966, 2974, 2998, 2999, 3011, 3019, 3022, 3046, 3067, 3079, 3083, 3086, 3118, 3119, 3142, 3167, 3187, 3191, 3203, 3214, 3254, 3259, 3271, 3299, 3307, 3319, 3323, 3326, 3331, 3334, 3343, 3347, 3359, 3371, 3398, 3407, 3446, 3463, 3467, 3491, 3494, 3499, 3511, 3518, 3527, 3539, 3547, 3559, 3566, 3571, 3583, 3607, 3622, 3623, 3631, 3643, 3646, 3659, 3662, 3671, 3691, 3694, 3727, 3734, 3742, 3758, 3767, 3779, 3823, 3847, 3851, 3862, 3863, 3902, 3907, 3911, 3919, 3923, 3931, 3943, 3947, 3974, 3998, 4003, 4006, 4007, 4019, 4022, 4027, 4051, 4054, 4078, 4079, 4091, 4099, 4111, 4126, 4127, 4166, 4174, 4198, 4211, 4219, 4222, 4231, 4243, 4259, 4262, 4327, 4339, 4363, 4391, 4406, 4414, 4423, 4447, 4451, 4463, 4478, 4483, 4486, 4502, 4507, 4519, 4523, 4547, 4567, 4574, 4583, 4603, 4639, 4643, 4663, 4678, 4679, 4691, 4694, 4703, 4723, 4742, 4751, 4766, 4783, 4787, 4798, 4799, 4831, 4846, 4871, 4894, 4903, 4918, 4919, 4931, 4943, 4951, 4967, 4987, 4999, 5003, 5006, 5023, 5051, 5059, 5062, 5078, 5086, 5087, 5119, 5147, 5158, 5167, 5171, 5179, 5227, 5231, 5279, 5294, 5318, 5323, 5326, 5342, 5347, 5351, 5366, 5374, 5387, 5398, 5399, 5407, 5414, 5419, 5422, 5438, 5443, 5462, 5471, 5479, 5483, 5507, 5519, 5531, 5534, 5563, 5582, 5591, 5606, 5638, 5639, 5647, 5651, 5659, 5686, 5702, 5758, 5779, 5783, 5791, 5806, 5807, 5839, 5843, 5851, 5854, 5867, 5879, 5923, 5926, 5939, 5942, 5987, 5998, 6007, 6011, 6022, 6038, 6043, 6046, 6047, 6067, 6079, 6091, 6131, 6134, 6143, 6158, 6163, 6166, 6199, 6203, 6211, 6247, 6263, 6271, 6287, 6299, 6323, 6326, 6343, 6359, 6367, 6374, 6379, 6382, 6406, 6451, 6491, 6502, 6518, 6542, 6547, 6551, 6599, 6607, 6614, 6619, 6638, 6646, 6659, 6662, 6679, 6691, 6694, 6703, 6718, 6719, 6742, 6763, 6803, 6814, 6823, 6827, 6863, 6871, 6883, 6899, 6907, 6911, 6926, 6934, 6947, 6959, 6967, 6971, 6982, 6983, 6991, 6998, 7022, 7027, 7039, 7043, 7078, 7079, 7094, 7103, 7118, 7127, 7142, 7151, 7159, 7166, 7187, 7207, 7211, 7214, 7219, 7243, 7246, 7247, 7283, 7286, 7307, 7318, 7342, 7351, 7382, 7411, 7438, 7451, 7454, 7478, 7487, 7499, 7507, 7523, 7534, 7547, 7558, 7559, 7583, 7603, 7606, 7607, 7646, 7687, 7691, 7699, 7702, 7703, 7723, 7727, 7759, 7814, 7823, 7838, 7867, 7886, 7894, 7907, 7927, 7934, 7951, 7963, 8006, 8011, 8014, 8038, 8039, 8054, 8087, 8111, 8123, 8147, 8158, 8167, 8171, 8179, 8182, 8191, 8198, 8219, 8222, 8231, 8243, 8254, 8263, 8278, 8287, 8291, 8318, 8387, 8419, 8422, 8423, 8431, 8443, 8467, 8486, 8527, 8542, 8563, 8566, 8599, 8627, 8654, 8663, 8678, 8699, 8719, 8726, 8747, 8779, 8783, 8807, 8819, 8831, 8839, 8846, 8867, 8887, 8894, 8902, 8923, 8926, 8951, 8963, 8966, 8971, 9007, 9011, 9014, 9038, 9043, 9046, 9059, 9067, 9091, 9103, 9127, 9134, 9182, 9187, 9199, 9203, 9206, 9227, 9239, 9283, 9286, 9302, 9311, 9319, 9343, 9358, 9391, 9403, 9406, 9439, 9446, 9463, 9491, 9502, 9511, 9518, 9547, 9566, 9574, 9587, 9598, 9619, 9623, 9631, 9643, 9662, 9679, 9719, 9739, 9743, 9767, 9787, 9791, 9806, 9811, 9838, 9839, 9851, 9859, 9862, 9883, 9887, 9923, 9931, 9934, 9967, 9974
類数 2 (1473)
$\operatorname{Cl}(K) \cong C_{2}$ (1473)
10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 66, 70, 74, 78, 87, 91, 95, 102, 106, 110, 111, 114, 115, 119, 122, 123, 138, 143, 146, 154, 155, 159, 174, 178, 182, 183, 186, 187, 190, 194, 202, 203, 215, 218, 222, 230, 238, 246, 247, 258, 259, 266, 267, 282, 286, 287, 295, 298, 299, 303, 310, 314, 318, 319, 327, 335, 339, 354, 355, 362, 366, 371, 374, 386, 391, 394, 395, 402, 403, 406, 407, 411, 415, 418, 426, 430, 447, 451, 458, 466, 470, 471, 474, 482, 494, 498, 511, 515, 518, 519, 527, 534, 535, 538, 543, 551, 554, 559, 562, 583, 586, 590, 591, 598, 602, 606, 611, 618, 623, 634, 635, 638, 642, 654, 655, 667, 670, 671, 678, 679, 682, 687, 695, 698, 699, 703, 707, 710, 742, 746, 755, 762, 763, 767, 771, 778, 779, 782, 790, 794, 802, 803, 806, 807, 814, 815, 822, 826, 830, 831, 834, 835, 843, 851, 854, 866, 871, 879, 902, 922, 923, 938, 942, 946, 951, 955, 978, 995, 1002, 1007, 1018, 1022, 1027, 1030, 1034, 1038, 1043, 1047, 1054, 1055, 1059, 1070, 1074, 1079, 1082, 1099, 1102, 1114, 1115, 1118, 1119, 1135, 1146, 1147, 1159, 1162, 1166, 1167, 1178, 1182, 1194, 1195, 1199, 1202, 1203, 1207, 1219, 1222, 1234, 1246, 1247, 1255, 1263, 1266, 1267, 1270, 1282, 1298, 1306, 1310, 1315, 1322, 1334, 1338, 1343, 1346, 1347, 1354, 1355, 1358, 1362, 1363, 1374, 1379, 1383, 1390, 1391, 1398, 1402, 1403, 1414, 1415, 1418, 1434, 1466, 1474, 1498, 1506, 1510, 1514, 1526, 1535, 1538, 1542, 1555, 1558, 1562, 1563, 1578, 1591, 1594, 1603, 1606, 1614, 1623, 1626, 1630, 1634, 1642, 1643, 1651, 1655, 1658, 1662, 1670, 1671, 1686, 1687, 1691, 1698, 1702, 1703, 1706, 1707, 1711, 1727, 1738, 1739, 1742, 1754, 1779, 1786, 1790, 1795, 1798, 1799, 1807, 1819, 1826, 1834, 1835, 1839, 1842, 1843, 1846, 1858, 1874, 1883, 1891, 1895, 1902, 1903, 1910, 1915, 1919, 1923, 1927, 1939, 1943, 1946, 1959, 1963, 1978, 1983, 1986, 1990, 1991, 1994, 2014, 2018, 2042, 2047, 2051, 2054, 2059, 2066, 2071, 2082, 2086, 2094, 2095, 2098, 2103, 2110, 2119, 2122, 2123, 2127, 2138, 2147, 2154, 2155, 2158, 2167, 2171, 2183, 2186, 2191, 2194, 2195, 2198, 2199, 2202, 2215, 2218, 2219, 2222, 2227, 2231, 2234, 2238, 2242, 2266, 2270, 2271, 2274, 2279, 2282, 2283, 2291, 2294, 2315, 2318, 2319, 2323, 2327, 2334, 2354, 2363, 2382, 2386, 2390, 2391, 2395, 2398, 2406, 2407, 2414, 2422, 2426, 2427, 2435, 2438, 2443, 2455, 2458, 2463, 2471, 2474, 2479, 2483, 2487, 2491, 2494, 2506, 2507, 2514, 2515, 2519, 2526, 2534, 2546, 2559, 2563, 2571, 2578, 2586, 2587, 2594, 2611, 2615, 2623, 2627, 2630, 2631, 2634, 2643, 2651, 2658, 2686, 2694, 2698, 2710, 2714, 2722, 2723, 2735, 2743, 2746, 2747, 2759, 2762, 2766, 2771, 2778, 2779, 2782, 2787, 2794, 2802, 2806, 2815, 2818, 2823, 2830, 2839, 2855, 2858, 2859, 2863, 2866, 2867, 2874, 2882, 2899, 2906, 2911, 2923, 2931, 2935, 2946, 2947, 2951, 2954, 2959, 2978, 2983, 2987, 2991, 2994, 2995, 3002, 3018, 3031, 3039, 3058, 3063, 3070, 3082, 3091, 3095, 3098, 3103, 3107, 3110, 3126, 3127, 3131, 3139, 3147, 3151, 3154, 3155, 3178, 3182, 3183, 3194, 3206, 3207, 3215, 3218, 3223, 3226, 3227, 3235, 3239, 3242, 3263, 3274, 3278, 3279, 3282, 3286, 3291, 3295, 3302, 3310, 3322, 3327, 3338, 3342, 3351, 3374, 3378, 3382, 3383, 3386, 3406, 3414, 3415, 3418, 3426, 3427, 3431, 3439, 3454, 3455, 3466, 3470, 3478, 3487, 3506, 3522, 3523, 3543, 3551, 3558, 3563, 3578, 3586, 3587, 3611, 3614, 3635, 3638, 3639, 3642, 3647, 3651, 3658, 3674, 3678, 3683, 3686, 3687, 3695, 3702, 3707, 3711, 3714, 3715, 3743, 3746, 3754, 3755, 3763, 3766, 3786, 3787, 3790, 3799, 3802, 3806, 3818, 3826, 3827, 3830, 3831, 3838, 3839, 3854, 3858, 3866, 3867, 3878, 3882, 3886, 3899, 3903, 3914, 3918, 3926, 3935, 3938, 3946, 3954, 3959, 3979, 3982, 3991, 3994, 4031, 4034, 4039, 4042, 4043, 4055, 4058, 4062, 4066, 4083, 4087, 4102, 4103, 4115, 4118, 4119, 4135, 4138, 4142, 4143, 4146, 4154, 4162, 4163, 4178, 4187, 4190, 4195, 4202, 4206, 4207, 4246, 4247, 4254, 4282, 4287, 4291, 4295, 4298, 4299, 4303, 4307, 4310, 4314, 4315, 4322, 4331, 4342, 4343, 4351, 4366, 4379, 4382, 4390, 4398, 4399, 4415, 4426, 4430, 4435, 4438, 4442, 4443, 4454, 4458, 4462, 4474, 4479, 4499, 4506, 4526, 4531, 4535, 4538, 4542, 4558, 4562, 4566, 4571, 4595, 4618, 4619, 4627, 4630, 4631, 4634, 4638, 4642, 4646, 4647, 4654, 4659, 4666, 4667, 4682, 4687, 4714, 4722, 4726, 4727, 4735, 4739, 4747, 4782, 4786, 4790, 4791, 4803, 4811, 4814, 4819, 4826, 4834, 4835, 4839, 4847, 4854, 4855, 4858, 4859, 4863, 4866, 4870, 4874, 4882, 4883, 4906, 4911, 4915, 4922, 4926, 4938, 4942, 4946, 4955, 4958, 4962, 4963, 4974, 4978, 4979, 4990, 5007, 5014, 5027, 5030, 5034, 5038, 5074, 5079, 5091, 5111, 5114, 5118, 5123, 5126, 5127, 5142, 5143, 5146, 5154, 5155, 5163, 5174, 5191, 5195, 5198, 5199, 5218, 5222, 5223, 5230, 5242, 5246, 5251, 5254, 5258, 5262, 5267, 5282, 5286, 5287, 5298, 5299, 5306, 5311, 5315, 5322, 5339, 5354, 5363, 5367, 5386, 5411, 5435, 5442, 5446, 5447, 5455, 5459, 5466, 5470, 5482, 5486, 5494, 5498, 5506, 5514, 5515, 5518, 5522, 5539, 5543, 5546, 5558, 5567, 5574, 5579, 5583, 5587, 5594, 5599, 5602, 5603, 5646, 5662, 5667, 5671, 5674, 5678, 5682, 5703, 5707, 5710, 5714, 5718, 5723, 5726, 5739, 5747, 5755, 5759, 5762, 5771, 5786, 5794, 5802, 5803, 5815, 5818, 5822, 5826, 5834, 5846, 5847, 5855, 5862, 5866, 5870, 5891, 5894, 5899, 5902, 5914, 5919, 5935, 5946, 5959, 5962, 5966, 5971, 5982, 5990, 5999, 6019, 6023, 6031, 6059, 6062, 6070, 6074, 6094, 6098, 6106, 6107, 6114, 6115, 6119, 6122, 6126, 6127, 6139, 6142, 6155, 6159, 6167, 6179, 6182, 6186, 6187, 6190, 6191, 6194, 6206, 6207, 6214, 6218, 6226, 6227, 6234, 6254, 6259, 6262, 6278, 6283, 6295, 6298, 6302, 6306, 6310, 6331, 6338, 6346, 6362, 6371, 6378, 6383, 6387, 6394, 6403, 6407, 6415, 6430, 6431, 6439, 6442, 6443, 6446, 6454, 6455, 6458, 6459, 6463, 6467, 6470, 6478, 6487, 6503, 6506, 6514, 6522, 6523, 6526, 6527, 6535, 6538, 6539, 6546, 6582, 6583, 6587, 6590, 6595, 6602, 6611, 6618, 6634, 6635, 6654, 6658, 6663, 6667, 6702, 6711, 6731, 6738, 6739, 6743, 6746, 6754, 6758, 6767, 6778, 6794, 6799, 6802, 6826, 6830, 6838, 6842, 6847, 6862, 6874, 6878, 6879, 6898, 6906, 6910, 6914, 6922, 6931, 6938, 6943, 6974, 6978, 6979, 6986, 6999, 7003, 7023, 7026, 7034, 7046, 7066, 7067, 7071, 7082, 7086, 7087, 7091, 7099, 7102, 7111, 7114, 7115, 7122, 7123, 7135, 7138, 7143, 7147, 7158, 7162, 7167, 7171, 7174, 7179, 7183, 7190, 7195, 7198, 7199, 7222, 7251, 7255, 7258, 7270, 7271, 7274, 7278, 7291, 7294, 7295, 7302, 7303, 7322, 7323, 7327, 7334, 7338, 7343, 7346, 7354, 7355, 7358, 7363, 7366, 7367, 7379, 7386, 7390, 7391, 7402, 7403, 7414, 7418, 7422, 7427, 7430, 7431, 7435, 7447, 7463, 7466, 7483, 7486, 7495, 7498, 7519, 7526, 7543, 7554, 7555, 7562, 7574, 7586, 7594, 7615, 7619, 7622, 7631, 7634, 7642, 7647, 7651, 7654, 7662, 7678, 7679, 7698, 7706, 7715, 7718, 7734, 7739, 7747, 7751, 7754, 7763, 7771, 7778, 7783, 7787, 7795, 7798, 7799, 7802, 7806, 7807, 7811, 7818, 7819, 7827, 7831, 7834, 7835, 7842, 7855, 7859, 7863, 7882, 7895, 7898, 7899, 7906, 7914, 7915, 7918, 7958, 7962, 7971, 7978, 7979, 7982, 7991, 8003, 8018, 8026, 8027, 8031, 8035, 8042, 8047, 8062, 8063, 8071, 8078, 8079, 8086, 8114, 8122, 8131, 8135, 8143, 8159, 8166, 8174, 8186, 8187, 8202, 8203, 8206, 8207, 8218, 8223, 8230, 8234, 8238, 8247, 8251, 8258, 8259, 8270, 8286, 8302, 8314, 8315, 8326, 8327, 8331, 8338, 8351, 8359, 8367, 8371, 8378, 8383, 8390, 8391, 8394, 8402, 8411, 8414, 8434, 8454, 8458, 8459, 8471, 8474, 8482, 8483, 8491, 8494, 8495, 8506, 8507, 8511, 8522, 8531, 8551, 8558, 8562, 8567, 8574, 8579, 8582, 8583, 8590, 8594, 8598, 8606, 8611, 8614, 8615, 8626, 8630, 8634, 8639, 8659, 8662, 8666, 8674, 8682, 8686, 8691, 8698, 8702, 8706, 8714, 8718, 8727, 8734, 8743, 8746, 8751, 8754, 8758, 8759, 8767, 8791, 8795, 8798, 8818, 8826, 8830, 8842, 8843, 8851, 8858, 8870, 8871, 8879, 8886, 8898, 8903, 8907, 8914, 8915, 8922, 8927, 8935, 8939, 8947, 8958, 8983, 8986, 8994, 8998, 9002, 9019, 9022, 9031, 9047, 9055, 9066, 9070, 9071, 9082, 9083, 9098, 9107, 9110, 9111, 9115, 9119, 9122, 9123, 9131, 9138, 9142, 9158, 9167, 9179, 9183, 9186, 9190, 9194, 9202, 9211, 9223, 9226, 9238, 9242, 9247, 9254, 9258, 9259, 9262, 9263, 9267, 9274, 9287, 9299, 9307, 9318, 9327, 9334, 9335, 9346, 9347, 9354, 9355, 9374, 9383, 9395, 9398, 9402, 9426, 9427, 9451, 9454, 9466, 9470, 9478, 9482, 9487, 9494, 9498, 9514, 9527, 9535, 9538, 9542, 9543, 9562, 9563, 9582, 9599, 9607, 9622, 9626, 9627, 9638, 9642, 9647, 9655, 9658, 9663, 9667, 9671, 9674, 9678, 9683, 9686, 9687, 9694, 9703, 9706, 9707, 9714, 9726, 9727, 9734, 9746, 9754, 9755, 9759, 9762, 9763, 9771, 9794, 9814, 9818, 9822, 9827, 9854, 9866, 9874, 9899, 9903, 9910, 9914, 9935, 9943, 9946, 9958, 9959, 9978, 9983, 9987, 9994
類数 3 (100)
$\operatorname{Cl}(K) \cong C_{3}$ (100)
79, 142, 223, 254, 326, 359, 443, 659, 839, 934, 1091, 1171, 1223, 1367, 1478, 1523, 1567, 1627, 1646, 1787, 1811, 1847, 1907, 1987, 2099, 2143, 2207, 2459, 2543, 2638, 2659, 2711, 2894, 2902, 2918, 2971, 3023, 3062, 3158, 3163, 3166, 3391, 3574, 3739, 3803, 3958, 4159, 4283, 4286, 4651, 4822, 4934, 5011, 5099, 5102, 5303, 5431, 5503, 5683, 5903, 6238, 6311, 6334, 6427, 6571, 6686, 6782, 6791, 7019, 7262, 7459, 7639, 7643, 7694, 7726, 7822, 7846, 7879, 7883, 8363, 8438, 8447, 8518, 8543, 8623, 8731, 8782, 8863, 8999, 9094, 9151, 9278, 9323, 9382, 9419, 9467, 9479, 9551, 9902, 9907
類数 4 (965)
$\operatorname{Cl}(K) \cong C_2 \times C_2$ (683)
130, 170, 195, 210, 231, 255, 290, 330, 370, 390, 410, 435, 455, 462, 483, 510, 530, 546, 555, 570, 595, 610, 615, 627, 651, 663, 690, 714, 715, 754, 759, 770, 795, 798, 858, 890, 903, 915, 930, 935, 962, 966, 970, 986, 987, 1010, 1015, 1066, 1095, 1110, 1130, 1131, 1190, 1218, 1230, 1235, 1254, 1258, 1295, 1302, 1311, 1326, 1330, 1335, 1370, 1378, 1394, 1407, 1410, 1419, 1430, 1435, 1455, 1463, 1479, 1482, 1490, 1491, 1515, 1518, 1547, 1551, 1554, 1570, 1586, 1595, 1610, 1615, 1635, 1659, 1695, 1722, 1730, 1767, 1770, 1794, 1802, 1810, 1830, 1855, 1870, 1887, 1898, 1930, 1938, 1955, 1970, 1974, 2002, 2015, 2030, 2046, 2067, 2074, 2090, 2091, 2130, 2135, 2146, 2163, 2190, 2226, 2235, 2247, 2255, 2262, 2290, 2314, 2330, 2343, 2346, 2355, 2370, 2378, 2387, 2418, 2431, 2442, 2470, 2478, 2522, 2530, 2555, 2562, 2570, 2590, 2595, 2607, 2618, 2622, 2626, 2639, 2667, 2670, 2679, 2690, 2715, 2770, 2795, 2814, 2834, 2838, 2847, 2870, 2886, 2895, 2910, 2926, 2930, 2938, 2955, 2958, 2967, 2982, 2990, 3010, 3034, 3055, 3059, 3066, 3074, 3090, 3094, 3102, 3111, 3115, 3130, 3162, 3170, 3171, 3190, 3198, 3219, 3230, 3270, 3290, 3311, 3318, 3335, 3354, 3355, 3367, 3370, 3390, 3395, 3410, 3423, 3435, 3451, 3458, 3471, 3486, 3490, 3495, 3498, 3507, 3515, 3530, 3534, 3535, 3538, 3542, 3562, 3567, 3619, 3655, 3666, 3710, 3723, 3730, 3731, 3738, 3759, 3774, 3783, 3810, 3815, 3835, 3842, 3855, 3874, 3890, 3895, 3910, 3922, 3930, 3939, 3995, 4002, 4010, 4015, 4026, 4030, 4035, 4047, 4070, 4071, 4074, 4082, 4090, 4110, 4123, 4130, 4134, 4147, 4155, 4170, 4179, 4182, 4186, 4191, 4210, 4218, 4234, 4242, 4255, 4278, 4326, 4330, 4346, 4370, 4371, 4386, 4395, 4403, 4407, 4431, 4466, 4494, 4495, 4498, 4503, 4510, 4514, 4522, 4530, 4539, 4543, 4551, 4570, 4578, 4587, 4602, 4610, 4611, 4615, 4683, 4686, 4690, 4695, 4710, 4730, 4731, 4746, 4758, 4774, 4794, 4795, 4818, 4823, 4862, 4879, 4947, 4983, 4991, 5010, 5015, 5018, 5019, 5035, 5055, 5066, 5083, 5090, 5106, 5110, 5122, 5151, 5159, 5162, 5170, 5214, 5215, 5226, 5235, 5270, 5278, 5291, 5295, 5307, 5334, 5335, 5338, 5343, 5358, 5370, 5394, 5395, 5406, 5410, 5423, 5430, 5451, 5467, 5487, 5495, 5502, 5510, 5523, 5530, 5538, 5551, 5555, 5559, 5570, 5590, 5642, 5658, 5690, 5691, 5694, 5695, 5698, 5719, 5727, 5730, 5735, 5754, 5763, 5770, 5790, 5810, 5811, 5830, 5835, 5838, 5858, 5863, 5871, 5874, 5882, 5883, 5890, 5934, 5955, 5970, 5995, 6015, 6018, 6035, 6055, 6063, 6095, 6118, 6130, 6154, 6170, 6215, 6222, 6230, 6231, 6235, 6251, 6258, 6266, 6303, 6314, 6315, 6322, 6330, 6335, 6342, 6351, 6391, 6438, 6466, 6474, 6479, 6490, 6495, 6519, 6530, 6531, 6554, 6578, 6586, 6603, 6610, 6643, 6690, 6695, 6698, 6710, 6715, 6730, 6734, 6735, 6747, 6755, 6790, 6815, 6846, 6851, 6855, 6870, 6882, 6895, 6902, 6915, 6919, 6923, 6935, 6942, 6951, 6954, 6955, 6963, 6987, 6990, 7010, 7015, 7030, 7090, 7106, 7130, 7134, 7163, 7170, 7178, 7194, 7202, 7210, 7238, 7239, 7266, 7287, 7298, 7306, 7314, 7347, 7370, 7383, 7446, 7458, 7462, 7467, 7474, 7490, 7503, 7518, 7527, 7530, 7535, 7539, 7579, 7599, 7602, 7611, 7618, 7626, 7630, 7635, 7667, 7670, 7683, 7707, 7710, 7714, 7730, 7738, 7790, 7810, 7815, 7826, 7843, 7878, 7887, 7890, 7910, 7931, 7946, 7955, 7959, 7970, 7990, 7998, 8007, 8015, 8022, 8030, 8034, 8043, 8058, 8066, 8138, 8155, 8162, 8170, 8178, 8195, 8210, 8215, 8239, 8242, 8246, 8290, 8310, 8319, 8323, 8346, 8355, 8358, 8362, 8382, 8395, 8430, 8435, 8439, 8455, 8502, 8535, 8554, 8570, 8602, 8607, 8642, 8679, 8687, 8690, 8695, 8723, 8755, 8762, 8763, 8770, 8787, 8810, 8814, 8823, 8827, 8862, 8866, 8895, 8911, 8938, 8943, 8990, 8995, 9039, 9042, 9051, 9074, 9078, 9095, 9102, 9106, 9130, 9139, 9146, 9159, 9170, 9174, 9178, 9191, 9195, 9230, 9231, 9246, 9255, 9303, 9330, 9339, 9366, 9367, 9370, 9390, 9399, 9415, 9418, 9443, 9447, 9455, 9462, 9483, 9499, 9515, 9519, 9530, 9534, 9546, 9579, 9591, 9614, 9615, 9635, 9646, 9695, 9698, 9699, 9715, 9723, 9730, 9758, 9770, 9786, 9815, 9831, 9842, 9843, 9858, 9879, 9890, 9894, 9906, 9911, 9915, 9919, 9930, 9951, 9962, 9966, 9970, 9982
$\operatorname{Cl}(K) \cong C_{4}$ (282)
82, 219, 274, 291, 322, 323, 434, 438, 514, 579, 582, 626, 658, 674, 706, 723, 731, 791, 818, 914, 939, 943, 959, 979, 1003, 1011, 1042, 1067, 1106, 1138, 1139, 1154, 1158, 1186, 1227, 1243, 1271, 1371, 1387, 1411, 1426, 1442, 1462, 1507, 1582, 1598, 1618, 1631, 1679, 1731, 1751, 1762, 1763, 1778, 1803, 1878, 1886, 1906, 1918, 1958, 1967, 2006, 2022, 2114, 2159, 2162, 2258, 2278, 2306, 2307, 2338, 2359, 2454, 2486, 2498, 2542, 2567, 2598, 2603, 2642, 2674, 2702, 2742, 2774, 2786, 2807, 2822, 2827, 3007, 3027, 3043, 3099, 3106, 3122, 3143, 3199, 3262, 3266, 3314, 3346, 3358, 3387, 3443, 3459, 3462, 3502, 3503, 3603, 3634, 3747, 3778, 3791, 3794, 3891, 3934, 3983, 3986, 4038, 4183, 4223, 4258, 4267, 4274, 4306, 4318, 4319, 4402, 4411, 4427, 4439, 4467, 4487, 4546, 4559, 4594, 4614, 4711, 4718, 4754, 4763, 4827, 4838, 4891, 4898, 5026, 5134, 5138, 5206, 5207, 5234, 5259, 5314, 5359, 5362, 5371, 5378, 5403, 5426, 5458, 5554, 5627, 5654, 5663, 5666, 5699, 5731, 5767, 5842, 5906, 5911, 5963, 5979, 5983, 6002, 6034, 6051, 6054, 6103, 6198, 6267, 6286, 6319, 6339, 6347, 6386, 6398, 6411, 6418, 6434, 6482, 6483, 6494, 6511, 6674, 6706, 6707, 6787, 6818, 6839, 6843, 6866, 6886, 6918, 7031, 7131, 7186, 7206, 7223, 7231, 7419, 7494, 7531, 7538, 7663, 7682, 7762, 7782, 7851, 7871, 7874, 7903, 7926, 7939, 7967, 8002, 8067, 8119, 8126, 8139, 8146, 8342, 8347, 8366, 8407, 8446, 8498, 8499, 8534, 8546, 8571, 8578, 8638, 8774, 8786, 8822, 8859, 8878, 8882, 8891, 8934, 8942, 9003, 9058, 9079, 9118, 9147, 9154, 9271, 9298, 9314, 9362, 9363, 9407, 9422, 9458, 9523, 9526, 9571, 9602, 9651, 9654, 9682, 9691, 9778, 9782, 9863, 9878, 9938, 9939, 9942, 9979, 9986
類数 5 (39)
$\operatorname{Cl}(K) \cong C_{5}$ (39)
439, 499, 727, 982, 1126, 1327, 1486, 1766, 2027, 2399, 2878, 3134, 3238, 3251, 3814, 3967, 4271, 4358, 4591, 4622, 5107, 5527, 5711, 5774, 5927, 6563, 6779, 7331, 7591, 7862, 8102, 8539, 8707, 8803, 9166, 9371, 9431, 9742, 9803
類数 6 (175)
$\operatorname{Cl}(K) \cong C_{6}$ (175)
235, 346, 427, 506, 574, 786, 842, 874, 894, 895, 898, 899, 906, 1086, 1191, 1211, 1339, 1342, 1406, 1527, 1546, 1639, 1735, 1758, 1851, 1866, 1882, 1954, 2118, 2230, 2263, 2298, 2495, 2510, 2554, 2666, 2678, 2726, 2758, 2831, 2922, 2962, 3035, 3047, 3054, 3071, 3138, 3287, 3394, 3419, 3422, 3442, 3482, 3514, 3579, 3590, 3594, 3602, 3667, 3670, 3846, 3859, 3898, 3962, 4094, 4098, 4106, 4151, 4227, 4238, 4279, 4294, 4362, 4367, 4434, 4511, 4670, 4699, 4771, 4778, 4886, 4907, 4910, 4927, 4966, 4982, 4994, 5063, 5095, 5131, 5178, 5255, 5263, 5302, 5327, 5383, 5542, 5578, 5722, 5738, 5798, 5799, 5898, 5918, 5938, 5991, 6026, 6082, 6243, 6366, 6414, 6423, 6499, 6515, 6558, 6623, 6639, 6751, 6766, 6807, 6819, 6835, 6887, 6927, 6995, 7058, 7126, 7234, 7235, 7279, 7311, 7319, 7374, 7387, 7415, 7423, 7471, 7598, 7655, 7658, 7671, 7711, 7870, 7994, 7999, 8051, 8227, 8266, 8267, 8279, 8306, 8335, 8339, 8374, 8399, 8403, 8426, 8503, 8519, 8538, 8603, 8735, 9034, 9062, 9155, 9218, 9294, 9578, 9659, 9710, 9718, 9766, 9830, 9926, 9995
類数 7 (18)
$\operatorname{Cl}(K) \cong C_{7}$ (18)
1087, 1294, 1934, 2251, 2302, 2467, 4139, 4702, 5039, 5743, 5827, 6151, 6598, 7919, 8311, 8462, 9326, 9539
類数 8 (262)
$\operatorname{Cl}(K) \cong C_2 \times C_2 \times C_2$ (63)
1155, 1995, 2210, 2310, 2415, 2730, 3003, 3135, 3255, 3315, 3570, 3770, 3795, 3927, 3990, 4290, 4515, 4930, 4935, 5115, 5187, 5330, 5610, 5655, 6090, 6195, 6270, 6279, 6290, 6510, 6555, 6630, 6699, 6783, 6970, 7035, 7095, 7215, 7410, 7455, 7590, 7735, 7755, 7770, 7854, 7930, 7995, 8151, 8211, 8547, 8610, 8715, 8778, 8855, 8970, 9010, 9030, 9282, 9435, 9471, 9570, 9690, 9870
$\operatorname{Cl}(K) \cong C_2 \times C_4$ (138)
399, 442, 870, 910, 1023, 1122, 1239, 1290, 1443, 1590, 1743, 1914, 1947, 2010, 2035, 2211, 2379, 2410, 2451, 2482, 2490, 2706, 2739, 2751, 2810, 2915, 2919, 3030, 3210, 3243, 3298, 3363, 3399, 3434, 3531, 3615, 3819, 3894, 3999, 4199, 4270, 4323, 4422, 4470, 4490, 4658, 4674, 4715, 4767, 4890, 4895, 4902, 5002, 5135, 5190, 5210, 5271, 5379, 5402, 5434, 5474, 5478, 5795, 5907, 5910, 5943, 5986, 6010, 6042, 6083, 6099, 6110, 6123, 6135, 6355, 6402, 6410, 6447, 6567, 6622, 6666, 6682, 6726, 6771, 6810, 7059, 7062, 7070, 7230, 7378, 7482, 7491, 7610, 7638, 7719, 7847, 7922, 7923, 8070, 8090, 8103, 8106, 8142, 8194, 8283, 8294, 8322, 8398, 8490, 8510, 8515, 8555, 8635, 8643, 8671, 8738, 8790, 8799, 8906, 8931, 8979, 8987, 9015, 9035, 9210, 9222, 9266, 9290, 9291, 9338, 9434, 9503, 9510, 9554, 9595, 9798, 9807, 9955
$\operatorname{Cl}(K) \cong C_{8}$ (61)
226, 646, 799, 994, 1299, 1351, 1446, 2019, 2134, 2402, 2434, 2811, 3014, 3202, 3247, 3403, 3526, 3554, 3682, 3883, 3963, 4063, 4226, 4387, 4607, 4738, 4971, 5183, 5186, 5219, 5614, 5622, 5947, 6014, 6086, 6146, 6242, 6274, 6671, 6683, 6722, 6774, 6946, 7051, 7522, 7563, 7582, 7766, 7779, 8083, 8299, 8354, 8386, 8683, 8834, 8962, 8974, 9023, 9143, 9442, 9991
類数 9 (6)
$\operatorname{Cl}(K) \cong C_{9}$ (6)
1654, 3719, 4534, 5623, 5878, 9998
類数 10 (58)
$\operatorname{Cl}(K) \cong C_{10}$ (58)
1111, 1226, 2031, 2335, 2362, 2602, 2986, 3246, 3379, 3598, 3599, 3722, 3782, 3966, 4334, 4359, 4555, 4582, 4586, 4843, 4867, 5071, 5611, 5615, 5630, 5631, 6071, 6078, 6087, 6202, 6239, 6294, 6395, 6574, 6854, 6891, 7282, 7339, 7394, 7439, 7571, 7674, 7746, 8023, 8074, 8479, 8651, 8794, 9235, 9322, 9411, 9474, 9611, 9670, 9722, 9731, 9847, 9895
類数 11 (2)
$\operatorname{Cl}(K) \cong C_{11}$ (2)
5182, 8059
類数 12 (86)
$\operatorname{Cl}(K) \cong C_{2} \times C_6$ (65)
730, 1090, 1495, 1599, 1771, 1806, 2055, 2139, 2170, 2635, 2703, 2755, 3706, 3955, 4011, 4215, 4355, 4623, 4706, 4755, 4970, 5511, 5595, 5930, 5954, 6058, 6162, 6307, 6594, 6670, 6770, 6798, 6834, 7014, 7055, 7107, 7242, 7259, 7310, 7566, 7567, 7690, 7743, 7954, 8115, 8130, 8255, 8274, 8282, 8418, 8466, 8530, 8655, 8742, 8806, 8890, 8930, 9086, 9087, 9215, 9219, 9394, 9590, 9618, 9795
$\operatorname{Cl}(K) \cong C_{12}$ (21)
1522, 1714, 2419, 2914, 3606, 4171, 4471, 4939, 5042, 5266, 5331, 5619, 6806, 7006, 7858, 8098, 8711, 8854, 9379, 9507, 9586
類数 13 (2)
$\operatorname{Cl}(K) \cong C_{13}$ (2)
4759, 8647
類数 14 (16)
$\operatorname{Cl}(K) \cong C_{14}$ (16)
1534, 2026, 2599, 3595, 3679, 3811, 4378, 4579, 5734, 5951, 5974, 7510, 7666, 8095, 8587, 8618
類数 15 (2)
$\operatorname{Cl}(K) \cong C_{15}$ (2)
9871, 9886
類数 16 (39)
$\operatorname{Cl}(K) \cong C_{2} \times C_2 \times C_4$ (12)
4810, 4830, 6006, 6890, 7315, 7395, 8295, 8463, 8835, 9490, 9735, 9867
$\operatorname{Cl}(K) \cong C_{2} \times C_8$ (18)
3306, 4251, 4807, 4899, 6486, 6562, 7359, 7511, 7786, 8094, 8646, 8710, 8815, 9430, 9779, 9823, 9834, 9835
$\operatorname{Cl}(K) \cong C_{4} \times C_4$ (1)
3026
$\operatorname{Cl}(K) \cong C_{16}$ (8)
4354, 6178, 6626, 7042, 7063, 7426, 7966, 9026
類数 17 (1)
$\operatorname{Cl}(K) \cong C_{17}$ (1)
7054
類数 18 (9)
$\operatorname{Cl}(K) \cong C_{18}$ (9)
4954, 5098, 6559, 6631, 7226, 7891, 8899, 9606, 9799
類数 20 (9)
$\operatorname{Cl}(K) \cong C_{2} \times C_{10}$ (8)
3970, 6994, 7330, 7570, 8099, 9006, 9331, 9410
$\operatorname{Cl}(K) \cong C_{20}$ (1)
9214
類数 22 (3)
$\operatorname{Cl}(K) \cong C_{22}$ (3)
4762, 7627, 8110
類数 24 (2)
$\operatorname{Cl}(K) \cong C_{2} \times C_{12}$ (1)
9790
$\operatorname{Cl}(K) \cong C_{24}$ (1)
9634
類数 28 (1)
$\operatorname{Cl}(K) \cong C_{2} \times C_{14}$ (1)
5626
2. $d$ が素数の場合のイデアル類群
素数 $d = p \leq 100000$ に対する実二次体 $K=\mathbb{Q}(\sqrt{p})$ のイデアル類群は以下のように分類される.
$p \equiv 1 \pmod{4}$の場合
類数 1 (3805)
$\operatorname{Cl}(K) = 1$ (3805)
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 233, 241, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 757, 769, 773, 797, 809, 821, 829, 853, 857, 877, 881, 929, 937, 941, 953, 977, 997, 1013, 1021, 1033, 1049, 1061, 1069, 1097, 1109, 1117, 1153, 1181, 1193, 1201, 1213, 1217, 1237, 1249, 1277, 1289, 1301, 1321, 1361, 1381, 1409, 1433, 1453, 1481, 1493, 1549, 1553, 1597, 1609, 1613, 1621, 1637, 1657, 1669, 1693, 1697, 1709, 1721, 1733, 1741, 1753, 1777, 1789, 1801, 1861, 1873, 1877, 1889, 1913, 1933, 1949, 1973, 1993, 1997, 2017, 2053, 2069, 2113, 2129, 2137, 2141, 2161, 2221, 2237, 2269, 2273, 2281, 2293, 2297, 2309, 2333, 2341, 2357, 2377, 2381, 2389, 2393, 2417, 2437, 2441, 2473, 2477, 2521, 2549, 2593, 2609, 2617, 2621, 2633, 2657, 2689, 2693, 2729, 2741, 2749, 2753, 2789, 2797, 2801, 2833, 2837, 2861, 2897, 2909, 2953, 2957, 2969, 3001, 3037, 3041, 3049, 3061, 3089, 3109, 3169, 3209, 3217, 3257, 3301, 3313, 3329, 3361, 3373, 3389, 3413, 3433, 3449, 3457, 3461, 3469, 3517, 3529, 3533, 3541, 3557, 3581, 3593, 3613, 3617, 3637, 3673, 3677, 3697, 3701, 3709, 3733, 3761, 3769, 3793, 3797, 3821, 3833, 3853, 3881, 3917, 3929, 3989, 4013, 4021, 4049, 4057, 4073, 4093, 4129, 4133, 4153, 4157, 4177, 4201, 4217, 4241, 4253, 4261, 4273, 4289, 4297, 4337, 4349, 4373, 4397, 4421, 4457, 4513, 4517, 4549, 4561, 4621, 4637, 4657, 4673, 4721, 4733, 4789, 4793, 4801, 4813, 4817, 4861, 4877, 4909, 4937, 4957, 4969, 4973, 4993, 5009, 5021, 5077, 5101, 5113, 5153, 5189, 5197, 5209, 5233, 5237, 5309, 5381, 5393, 5413, 5437, 5441, 5449, 5501, 5557, 5569, 5573, 5581, 5641, 5653, 5657, 5669, 5689, 5693, 5701, 5717, 5737, 5749, 5801, 5813, 5849, 5857, 5861, 5869, 5881, 5897, 5953, 5981, 6029, 6037, 6073, 6089, 6101, 6121, 6173, 6197, 6217, 6221, 6229, 6257, 6269, 6277, 6301, 6317, 6329, 6337, 6353, 6361, 6373, 6389, 6397, 6421, 6449, 6469, 6473, 6521, 6529, 6553, 6569, 6577, 6581, 6653, 6661, 6673, 6689, 6701, 6709, 6733, 6737, 6761, 6781, 6793, 6829, 6833, 6841, 6857, 6869, 6917, 6961, 6977, 7001, 7013, 7069, 7109, 7121, 7129, 7177, 7193, 7213, 7237, 7253, 7297, 7309, 7321, 7333, 7349, 7369, 7393, 7417, 7433, 7457, 7477, 7489, 7517, 7529, 7541, 7549, 7561, 7577, 7589, 7621, 7649, 7669, 7681, 7717, 7741, 7757, 7789, 7793, 7829, 7841, 7853, 7877, 7901, 7933, 7937, 7949, 7993, 8009, 8053, 8081, 8089, 8093, 8117, 8161, 8209, 8221, 8233, 8237, 8269, 8273, 8293, 8297, 8317, 8329, 8353, 8369, 8377, 8389, 8429, 8461, 8513, 8521, 8537, 8573, 8609, 8629, 8641, 8669, 8677, 8681, 8693, 8737, 8741, 8753, 8821, 8849, 8861, 8893, 8929, 8933, 8941, 8969, 9001, 9013, 9041, 9109, 9137, 9157, 9161, 9173, 9209, 9221, 9241, 9257, 9277, 9337, 9341, 9349, 9377, 9397, 9421, 9433, 9437, 9461, 9473, 9497, 9521, 9533, 9601, 9613, 9629, 9649, 9661, 9677, 9689, 9697, 9721, 9733, 9769, 9781, 9817, 9857, 9901, 9929, 9941, 9949, 9973, 10009, 10037, 10061, 10093, 10133, 10141, 10169, 10177, 10181, 10193, 10253, 10289, 10321, 10337, 10369, 10429, 10433, 10453, 10477, 10501, 10513, 10529, 10589, 10597, 10601, 10657, 10709, 10729, 10753, 10781, 10789, 10837, 10853, 10861, 10937, 10973, 10993, 11069, 11093, 11113, 11117, 11149, 11161, 11173, 11177, 11213, 11261, 11317, 11329, 11353, 11369, 11393, 11437, 11489, 11497, 11549, 11593, 11597, 11617, 11633, 11657, 11677, 11681, 11689, 11701, 11717, 11801, 11813, 11833, 11897, 11909, 11933, 11941, 11953, 11969, 11981, 12037, 12049, 12073, 12097, 12109, 12113, 12149, 12157, 12241, 12253, 12277, 12289, 12329, 12373, 12377, 12413, 12421, 12433, 12437, 12457, 12473, 12517, 12541, 12553, 12569, 12601, 12613, 12637, 12641, 12653, 12689, 12697, 12713, 12721, 12757, 12781, 12809, 12829, 12841, 12853, 12889, 12893, 12917, 12941, 12953, 12973, 13001, 13009, 13037, 13093, 13109, 13121, 13177, 13217, 13241, 13249, 13297, 13309, 13337, 13381, 13397, 13417, 13421, 13441, 13469, 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66361, 66373, 66377, 66413, 66449, 66457, 66509, 66533, 66541, 66553, 66569, 66593, 66601, 66617, 66629, 66697, 66701, 66713, 66721, 66733, 66797, 66809, 66821, 66841, 66853, 66877, 66977, 67021, 67033, 67049, 67057, 67061, 67121, 67129, 67141, 67153, 67157, 67169, 67181, 67189, 67213, 67261, 67273, 67289, 67349, 67369, 67421, 67429, 67433, 67453, 67477, 67481, 67489, 67537, 67577, 67757, 67777, 67801, 67853, 67901, 67933, 67957, 67961, 68041, 68053, 68113, 68141, 68161, 68209, 68213, 68261, 68329, 68389, 68437, 68473, 68477, 68501, 68521, 68581, 68633, 68713, 68737, 68777, 68821, 68897, 68909, 68917, 68993, 69001, 69029, 69073, 69109, 69149, 69197, 69221, 69233, 69257, 69313, 69317, 69337, 69341, 69389, 69457, 69473, 69497, 69557, 69593, 69653, 69661, 69677, 69709, 69737, 69809, 69829, 69833, 69857, 69877, 69929, 69941, 70001, 70009, 70117, 70121, 70141, 70157, 70181, 70237, 70241, 70249, 70289, 70297, 70309, 70321, 70373, 70381, 70393, 70489, 70501, 70537, 70549, 70573, 70589, 70657, 70709, 70717, 70753, 70769, 70793, 70841, 70849, 70853, 70877, 70901, 70913, 70937, 70949, 70957, 70981, 70997, 71069, 71081, 71153, 71209, 71233, 71249, 71257, 71261, 71317, 71341, 71353, 71389, 71413, 71437, 71453, 71473, 71537, 71549, 71593, 71597, 71633, 71693, 71713, 71761, 71777, 71809, 71821, 71837, 71881, 71909, 71941, 71993, 72053, 72073, 72077, 72089, 72101, 72161, 72173, 72221, 72229, 72269, 72277, 72313, 72337, 72341, 72353, 72421, 72461, 72493, 72497, 72533, 72613, 72617, 72649, 72661, 72673, 72689, 72701, 72797, 72869, 72889, 72893, 72937, 72949, 72973, 72977, 73009, 73037, 73061, 73121, 73133, 73141, 73181, 73237, 73309, 73361, 73369, 73417, 73421, 73433, 73453, 73477, 73517, 73529, 73561, 73589, 73597, 73609, 73613, 73637, 73693, 73709, 73721, 73849, 73877, 73897, 73961, 73973, 74017, 74021, 74077, 74149, 74161, 74177, 74189, 74197, 74201, 74209, 74257, 74297, 74357, 74377, 74381, 74413, 74441, 74453, 74489, 74509, 74521, 74573, 74609, 74653, 74717, 74729, 74761, 74797, 74821, 74861, 74869, 74873, 74929, 74941, 75013, 75017, 75029, 75037, 75041, 75109, 75133, 75149, 75161, 75169, 75193, 75209, 75217, 75253, 75269, 75277, 75329, 75337, 75353, 75377, 75437, 75521, 75533, 75553, 75557, 75577, 75641, 75689, 75721, 75781, 75793, 75797, 75833, 75853, 75913, 75937, 75941, 75989, 75997, 76129, 76157, 76213, 76249, 76253, 76261, 76289, 76333, 76369, 76421, 76481, 76493, 76537, 76541, 76649, 76673, 76697, 76717, 76753, 76757, 76777, 76781, 76801, 76829, 76837, 76873, 76949, 76961, 77017, 77029, 77041, 77081, 77093, 77101, 77141, 77153, 77201, 77213, 77249, 77261, 77269, 77317, 77417, 77477, 77489, 77509, 77549, 77557, 77569, 77573, 77621, 77689, 77713, 77761, 77773, 77797, 77801, 77813, 77893, 77929, 77933, 77969, 77977, 78017, 78049, 78101, 78121, 78137, 78157, 78173, 78193, 78229, 78233, 78241, 78277, 78301, 78317, 78437, 78509, 78517, 78541, 78553, 78569, 78577, 78593, 78649, 78653, 78697, 78737, 78797, 78809, 78857, 78893, 78901, 78929, 78941, 78977, 78989, 79153, 79181, 79193, 79201, 79229, 79241, 79273, 79301, 79309, 79337, 79349, 79393, 79493, 79537, 79549, 79561, 79589, 79601, 79609, 79613, 79621, 79657, 79697, 79769, 79777, 79801, 79813, 79817, 79829, 79841, 79861, 79873, 79889, 79901, 79973, 80021, 80077, 80141, 80149, 80153, 80173, 80177, 80209, 80233, 80273, 80309, 80317, 80329, 80369, 80449, 80473, 80489, 80513, 80557, 80669, 80677, 80713, 80737, 80761, 80777, 80789, 80809, 80833, 80849, 80897, 80909, 80917, 80929, 80933, 80953, 81001, 81013, 81017, 81097, 81101, 81157, 81173, 81181, 81197, 81233, 81293, 81349, 81353, 81401, 81421, 81457, 81509, 81517, 81533, 81553, 81569, 81637, 81649, 81677, 81689, 81737, 81749, 81761, 81773, 81817, 81869, 81901, 81929, 81953, 81973, 82009, 82013, 82021, 82037, 82073, 82129, 82141, 82153, 82189, 82193, 82237, 82241, 82261, 82301, 82349, 82361, 82393, 82421, 82457, 82469, 82493, 82549, 82561, 82601, 82609, 82613, 82633, 82729, 82757, 82781, 82793, 82813, 82837, 82913, 82981, 82997, 83009, 83089, 83093, 83101, 83117, 83137, 83233, 83257, 83269, 83357, 83401, 83417, 83437, 83449, 83477, 83497, 83537, 83557, 83561, 83597, 83609, 83617, 83621, 83641, 83701, 83717, 83737, 83761, 83773, 83813, 83833, 83869, 83873, 83921, 84017, 84053, 84089, 84121, 84137, 84181, 84221, 84229, 84313, 84317, 84349, 84377, 84389, 84401, 84437, 84457, 84481, 84509, 84521, 84533, 84589, 84629, 84649, 84653, 84673, 84697, 84713, 84737, 84761, 84793, 84857, 84869, 84913, 84961, 84977, 85009, 85037, 85049, 85061, 85081, 85093, 85109, 85133, 85193, 85201, 85213, 85229, 85237, 85297, 85333, 85369, 85429, 85469, 85513, 85549, 85577, 85601, 85661, 85669, 85717, 85781, 85793, 85829, 85837, 85889, 85909, 85933, 86029, 86069, 86077, 86117, 86137, 86197, 86209, 86249, 86257, 86269, 86293, 86297, 86341, 86357, 86369, 86381, 86389, 86413, 86453, 86461, 86477, 86509, 86533, 86561, 86573, 86677, 86729, 86753, 86813, 86857, 86861, 86929, 86969, 86981, 87037, 87049, 87121, 87253, 87277, 87281, 87293, 87313, 87317, 87421, 87433, 87481, 87509, 87517, 87541, 87553, 87557, 87589, 87613, 87641, 87649, 87697, 87701, 87721, 87793, 87797, 87833, 87869, 87877, 87881, 87917, 87961, 88001, 88037, 88069, 88093, 88117, 88129, 88177, 88237, 88241, 88261, 88301, 88321, 88397, 88469, 88493, 88513, 88589, 88609, 88657, 88681, 88721, 88789, 88793, 88801, 88813, 88853, 88861, 88873, 88937, 88993, 88997, 89009, 89021, 89041, 89057, 89101, 89113, 89137, 89153, 89189, 89209, 89213, 89237, 89261, 89269, 89273, 89293, 89317, 89381, 89393, 89449, 89477, 89501, 89513, 89521, 89533, 89561, 89597, 89633, 89653, 89669, 89689, 89797, 89809, 89821, 89833, 89849, 89897, 89909, 89917, 89977, 89989, 90053, 90073, 90089, 90121, 90149, 90173, 90197, 90217, 90281, 90353, 90373, 90397, 90401, 90469, 90529, 90533, 90617, 90641, 90677, 90697, 90709, 90749, 90793, 90821, 90833, 90841, 90901, 90917, 90997, 91009, 91033, 91081, 91097, 91121, 91129, 91141, 91153, 91193, 91229, 91249, 91297, 91309, 91369, 91373, 91381, 91393, 91397, 91453, 91457, 91493, 91513, 91541, 91573, 91577, 91621, 91673, 91733, 91753, 91757, 91873, 91909, 91921, 91961, 91997, 92009, 92041, 92077, 92153, 92177, 92221, 92233, 92237, 92317, 92333, 92353, 92357, 92369, 92377, 92413, 92489, 92593, 92641, 92669, 92681, 92693, 92717, 92737, 92753, 92761, 92789, 92801, 92821, 92849, 92893, 92921, 92941, 92957, 93001, 93053, 93089, 93097, 93113, 93133, 93169, 93229, 93253, 93257, 93329, 93377, 93481, 93493, 93497, 93529, 93553, 93557, 93581, 93629, 93701, 93761, 93809, 93889, 93893, 93901, 93913, 93937, 93941, 93997, 94009, 94033, 94049, 94057, 94121, 94153, 94169, 94201, 94229, 94261, 94273, 94309, 94349, 94433, 94477, 94513, 94529, 94541, 94573, 94597, 94613, 94649, 94709, 94777, 94781, 94789, 94793, 94837, 94841, 94849, 94873, 94889, 94933, 94949, 94993, 95021, 95089, 95093, 95101, 95153, 95177, 95189, 95233, 95261, 95273, 95401, 95413, 95429, 95441, 95549, 95561, 95569, 95581, 95597, 95617, 95629, 95633, 95701, 95713, 95717, 95737, 95789, 95801, 95813, 95857, 95869, 95873, 95881, 95929, 95989, 96001, 96013, 96017, 96053, 96097, 96137, 96157, 96181, 96221, 96269, 96281, 96289, 96293, 96329, 96353, 96377, 96401, 96461, 96493, 96517, 96553, 96557, 96581, 96601, 96661, 96697, 96737, 96749, 96757, 96769, 96797, 96821, 96857, 96953, 96973, 96989, 97021, 97073, 97081, 97117, 97157, 97169, 97177, 97213, 97301, 97369, 97373, 97381, 97397, 97429, 97441, 97453, 97501, 97549, 97561, 97577, 97613, 97649, 97673, 97729, 97777, 97789, 97829, 97961, 98009, 98017, 98041, 98057, 98081, 98101, 98129, 98213, 98221, 98257, 98269, 98297, 98317, 98321, 98369, 98377, 98429, 98453, 98473, 98533, 98561, 98621, 98641, 98669, 98689, 98713, 98717, 98729, 98773, 98801, 98837, 98849, 98869, 98873, 98893, 98897, 98909, 98929, 98953, 98981, 98993, 99013, 99017, 99041, 99053, 99089, 99109, 99133, 99149, 99173, 99181, 99233, 99241, 99277, 99289, 99317, 99349, 99377, 99409, 99469, 99497, 99529, 99577, 99581, 99661, 99689, 99709, 99721, 99733, 99761, 99793, 99809, 99829, 99877, 99881, 99929, 99989
類数 3 (507)
$\operatorname{Cl}(K) \cong C_3$ (507)
229, 257, 733, 761, 1229, 1373, 1489, 1901, 2089, 2213, 2557, 2677, 2713, 2777, 2857, 2917, 3221, 3229, 3877, 3889, 4001, 4481, 4493, 4597, 4649, 4729, 4933, 5081, 5261, 5281, 5297, 5333, 5477, 5741, 5821, 6053, 6133, 6637, 6997, 7481, 7537, 7673, 7753, 8017, 8069, 8581, 8597, 8713, 8837, 9133, 9281, 9293, 9413, 9749, 9833, 10069, 10301, 10333, 10457, 10733, 10889, 10949, 11057, 11197, 11777, 11789, 11821, 12197, 12269, 12401, 12577, 12821, 13537, 13577, 13877, 14197, 14389, 14653, 14969, 15061, 15193, 15349, 15641, 15661, 15733, 15737, 15773, 15881, 16141, 16369, 16477, 16481, 16553, 16649, 16661, 16673, 17053, 17417, 17477, 17581, 17609, 17929, 18097, 18257, 18269, 18521, 18541, 18661, 18701, 19213, 19429, 19469, 19949, 20249, 20297, 20341, 20353, 20521, 20549, 20693, 20749, 21101, 21557, 21589, 21713, 21737, 22229, 22397, 22709, 22717, 22721, 22769, 23297, 23321, 23417, 23497, 23689, 23909, 23917, 23957, 23993, 24001, 24197, 24281, 24413, 24533, 24749, 24917, 25301, 25409, 25469, 25537, 25717, 25793, 25981, 26029, 26113, 26489, 26821, 26893, 27329, 27409, 27437, 27617, 27673, 28097, 28309, 28349, 28657, 28669, 28837, 29221, 29569, 29629, 29989, 30169, 30553, 30677, 30829, 30941, 31069, 31121, 31513, 31793, 31973, 32353, 32413, 32569, 32833, 33301, 33457, 33521, 33581, 33641, 34157, 34253, 34721, 34877, 35129, 35317, 35393, 35401, 35537, 35597, 36013, 36389, 36469, 36653, 36677, 36761, 37489, 37537, 38053, 38593, 38609, 38921, 38953, 38993, 39157, 39233, 39397, 39569, 39733, 40093, 41057, 41269, 41381, 41389, 41597, 41813, 42073, 42349, 42649, 42709, 42953, 43321, 43441, 43577, 43717, 43961, 44273, 44357, 44617, 44621, 44641, 45361, 45433, 45541, 45641, 45841, 46489, 46589, 46757, 47353, 47701, 47809, 47977, 48481, 48677, 48953, 49481, 49681, 49757, 50033, 50053, 50221, 50329, 50593, 50753, 51437, 51481, 51637, 52081, 52321, 52697, 52709, 52837, 53269, 53617, 53633, 54181, 54409, 54437, 54449, 54469, 54877, 55049, 55073, 55661, 55721, 55813, 55901, 56237, 56333, 56489, 56713, 57077, 57241, 57373, 58049, 58061, 58193, 58217, 58537, 59093, 59233, 59809, 60037, 60169, 60209, 60901, 61861, 61909, 62233, 62473, 62549, 62929, 63317, 63377, 63589, 63629, 64033, 64189, 64373, 64489, 64709, 64877, 64901, 65033, 65101, 65309, 65381, 65981, 66089, 66221, 66293, 66529, 66749, 66973, 67073, 67493, 67709, 67733, 67789, 68489, 68669, 68813, 69061, 69401, 69493, 69821, 69997, 70061, 70201, 70313, 70457, 70481, 70529, 70621, 70729, 70921, 71089, 71237, 71329, 71333, 71429, 71741, 71789, 71861, 71933, 72169, 72253, 72577, 72733, 72817, 73013, 73673, 73757, 74293, 74449, 74713, 74897, 74933, 75181, 75289, 75389, 75541, 75617, 75653, 75709, 75821, 75869, 76081, 76913, 77069, 77237, 77513, 77617, 77681, 77849, 78041, 78341, 78497, 78721, 78877, 79333, 79357, 79397, 79481, 79669, 79693, 79757, 80221, 80629, 80681, 80989, 81049, 81077, 81373, 81409, 81629, 81937, 82657, 83177, 83221, 83777, 83933, 84061, 84449, 84701, 85021, 85313, 85517, 85597, 85621, 85817, 86113, 86353, 86441, 86501, 86629, 86693, 86837, 86993, 87013, 87041, 87133, 87149, 87181, 87337, 87973, 88169, 88337, 88661, 88817, 88897, 89069, 89413, 89417, 89657, 89681, 89753, 90017, 90437, 90977, 90989, 91237, 91253, 91433, 91781, 91801, 91837, 92269, 92297, 92461, 92557, 92569, 92581, 92809, 92857, 92861, 92993, 93077, 93241, 93281, 93949, 94109, 94397, 94693, 95257, 95369, 95621, 96233, 96457, 96469, 96589, 97001, 97241, 97553, 97849, 98573, 98809, 99257, 99397, 99713, 99817, 99961
類数 5 (176)
$\operatorname{Cl}(K) \cong C_{5}$ (176)
401, 1093, 1429, 2081, 2153, 3121, 3181, 3253, 4357, 4441, 4889, 6113, 6481, 6949, 7229, 7817, 8501, 8689, 9181, 9829, 10613, 10909, 11273, 12041, 12101, 12301, 13229, 13841, 14281, 15121, 15473, 15889, 16001, 16741, 17033, 18253, 18353, 18433, 18773, 19477, 20089, 20389, 21317, 22621, 23173, 23677, 23801, 24593, 25037, 25097, 25153, 25621, 26309, 27749, 27893, 30341, 30893, 30949, 32609, 33029, 33317, 33797, 34361, 35509, 35593, 35933, 36541, 36929, 37253, 37781, 38177, 38333, 38561, 38629, 40637, 41453, 41777, 42157, 42437, 43037, 43889, 44741, 45533, 45553, 45989, 46337, 46933, 47857, 48437, 49081, 50909, 50969, 52313, 53437, 53441, 53993, 54881, 55201, 55457, 56101, 56681, 57457, 57649, 57809, 57853, 58321, 58657, 58897, 58921, 59149, 59393, 61493, 62213, 62873, 63697, 63761, 64849, 66653, 66949, 67589, 67993, 68597, 70177, 72469, 72953, 72997, 73277, 73553, 74093, 74317, 74561, 74597, 75773, 76001, 76597, 77377, 78713, 79133, 79997, 80537, 80621, 80701, 80749, 81041, 81281, 82529, 83341, 83389, 83653, 83857, 84421, 84809, 85361, 85453, 87257, 87473, 87629, 88729, 89017, 90473, 90481, 91841, 92381, 92401, 92657, 93337, 94117, 94621, 95393, 95461, 96149, 96893, 96997, 97841, 99137, 99833
類数 7 (95)
$\operatorname{Cl}(K) \cong C_{7}$ (95)
577, 1009, 1601, 2029, 4229, 5273, 5417, 9029, 9049, 10313, 10357, 10957, 11621, 12497, 13049, 13313, 15277, 15809, 16217, 16901, 17257, 17569, 17957, 19081, 22921, 23629, 23857, 26317, 26713, 29753, 30817, 31321, 33941, 35081, 35221, 35869, 39089, 39301, 40009, 40697, 41149, 41213, 42533, 46817, 46829, 49333, 51137, 51893, 52009, 52301, 53309, 53717, 53777, 55589, 55837, 56629, 57389, 58477, 60601, 62189, 63853, 65413, 65657, 65729, 67217, 68449, 69481, 71569, 71849, 71917, 72109, 73189, 76561, 76733, 77369, 77641, 79433, 80429, 81701, 81853, 82217, 85121, 85733, 86689, 87221, 89329, 92189, 93601, 94421, 94561, 95213, 95773, 96497, 97813, 98389
類数 9 (63)
$\operatorname{Cl}(K) \cong C_{3} \times C_{3}$ (2)
32009, 62501
$\operatorname{Cl}(K) \cong C_{9}$ (61)
1129, 3137, 4409, 5521, 7573, 7873, 10273, 12409, 15629, 17989, 19441, 24029, 24169, 24697, 24781, 26573, 28901, 33493, 38917, 38977, 40361, 40709, 46649, 47093, 47681, 48121, 48973, 49109, 53101, 53917, 54293, 58189, 59209, 60337, 60397, 61961, 62137, 64013, 67741, 67829, 71161, 74353, 77137, 77521, 78781, 78853, 80341, 81769, 83273, 86161, 87977, 91529, 91957, 91969, 94253, 94441, 95317, 95917, 97609, 97861, 99901
類数 11 (28)
$\operatorname{Cl}(K) \cong C_{11}$ (28)
1297, 12161, 12281, 12589, 14009, 16333, 20369, 22129, 22501, 36709, 43753, 57713, 57781, 65353, 66889, 71129, 73681, 74101, 78889, 82721, 83689, 85853, 86017, 86869, 88741, 94961, 98597, 98737
類数 13 (27)
$\operatorname{Cl}(K) \cong C_{13}$ (27)
8101, 13033, 13457, 15377, 15877, 21613, 31193, 33857, 41801, 45329, 46229, 49201, 49297, 58661, 60029, 65701, 68881, 75401, 76441, 79633, 82373, 83969, 87853, 88969, 90313, 95957, 97973
類数 15 (18)
$\operatorname{Cl}(K) \cong C_{15}$ (18)
11321, 13693, 17737, 18329, 25741, 35977, 37993, 42853, 51257, 52901, 53017, 53401, 55229, 69193, 71293, 72481, 88289, 92033
類数 17 (9)
$\operatorname{Cl}(K) \cong C_{17}$ (9)
11257, 30137, 42937, 47657, 54541, 55697, 58913, 64081, 65633
類数 19 (10)
$\operatorname{Cl}(K) \cong C_{19}$ (10)
18229, 31333, 38569, 44101, 49393, 68281, 70229, 70429, 90289, 93637
類数 21 (7)
$\operatorname{Cl}(K) \cong C_{21}$ (7)
7057, 49033, 65537, 69761, 75629, 82889, 83077
類数 23 (4)
$\operatorname{Cl}(K) \cong C_{23}$ (4)
23593, 30977, 67409, 91813
類数 25 (10)
$\operatorname{Cl}(K) \cong C_{25}$ (10)
27689, 50177, 59053, 59341, 68729, 68749, 80657, 85381, 92173, 95009
類数 27 (4)
$\operatorname{Cl}(K) \cong C_{27}$ (4)
8761, 21433, 36073, 65029
類数 29 (4)
$\operatorname{Cl}(K) \cong C_{29}$ (4)
56857, 72901, 86201, 99401
類数 33 (2)
$\operatorname{Cl}(K) \cong C_{33}$ (2)
63361, 74857
類数 35 (2)
$\operatorname{Cl}(K) \cong C_{35}$ (2)
25601, 96337
類数 37 (2)
$\operatorname{Cl}(K) \cong C_{37}$ (2)
24337, 94321
類数 39 (1)
$\operatorname{Cl}(K) \cong C_{39}$ (1)
55441
類数 43 (1)
$\operatorname{Cl}(K) \cong C_{43}$ (1)
14401
類数 45 (2)
$\operatorname{Cl}(K) \cong C_{45}$ (2)
32401, 67601
類数 47 (1)
$\operatorname{Cl}(K) \cong C_{47}$ (1)
78401
類数 49 (1)
$\operatorname{Cl}(K) \cong C_{49}$ (1)
70969
類数 51 (1)
$\operatorname{Cl}(K) \cong C_{51}$ (1)
69697
類数 57 (1)
$\operatorname{Cl}(K) \cong C_{57}$ (1)
41617
類数 63 (1)
$\operatorname{Cl}(K) \cong C_{63}$ (1)
57601
類数 87 (1)
$\operatorname{Cl}(K) \cong C_{87}$ (1)
90001
$p \equiv 2, 3 \pmod{4}$の場合
類数 1 (3842)
$\operatorname{Cl}(K) = 1$ (3842)
2, 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 367, 379, 383, 419, 431, 463, 467, 479, 487, 491, 503, 523, 547, 563, 571, 587, 599, 607, 619, 631, 643, 647, 683, 691, 719, 739, 743, 751, 787, 811, 823, 827, 859, 863, 883, 887, 907, 911, 919, 947, 967, 971, 983, 991, 1019, 1031, 1039, 1051, 1063, 1103, 1123, 1151, 1163, 1187, 1231, 1259, 1279, 1283, 1291, 1303, 1307, 1319, 1399, 1423, 1427, 1439, 1447, 1451, 1459, 1471, 1483, 1487, 1499, 1511, 1531, 1543, 1559, 1571, 1579, 1583, 1607, 1619, 1663, 1667, 1699, 1723, 1747, 1759, 1783, 1823, 1831, 1867, 1871, 1879, 1931, 1951, 1979, 1999, 2003, 2011, 2039, 2063, 2083, 2087, 2111, 2131, 2179, 2203, 2239, 2243, 2267, 2287, 2311, 2339, 2347, 2351, 2371, 2383, 2411, 2423, 2447, 2503, 2531, 2539, 2551, 2579, 2591, 2647, 2663, 2671, 2683, 2687, 2699, 2707, 2719, 2731, 2767, 2791, 2803, 2819, 2843, 2851, 2879, 2887, 2903, 2927, 2939, 2963, 2999, 3011, 3019, 3067, 3079, 3083, 3119, 3167, 3187, 3191, 3203, 3259, 3271, 3299, 3307, 3319, 3323, 3331, 3343, 3347, 3359, 3371, 3407, 3463, 3467, 3491, 3499, 3511, 3527, 3539, 3547, 3559, 3571, 3583, 3607, 3623, 3631, 3643, 3659, 3671, 3691, 3727, 3767, 3779, 3823, 3847, 3851, 3863, 3907, 3911, 3919, 3923, 3931, 3943, 3947, 4003, 4007, 4019, 4027, 4051, 4079, 4091, 4099, 4111, 4127, 4211, 4219, 4231, 4243, 4259, 4327, 4339, 4363, 4391, 4423, 4447, 4451, 4463, 4483, 4507, 4519, 4523, 4547, 4567, 4583, 4603, 4639, 4643, 4663, 4679, 4691, 4703, 4723, 4751, 4783, 4787, 4799, 4831, 4871, 4903, 4919, 4931, 4943, 4951, 4967, 4987, 4999, 5003, 5023, 5051, 5059, 5087, 5119, 5147, 5167, 5171, 5179, 5227, 5231, 5279, 5323, 5347, 5351, 5387, 5399, 5407, 5419, 5443, 5471, 5479, 5483, 5507, 5519, 5531, 5563, 5591, 5639, 5647, 5651, 5659, 5779, 5783, 5791, 5807, 5839, 5843, 5851, 5867, 5879, 5923, 5939, 5987, 6007, 6011, 6043, 6047, 6067, 6079, 6091, 6131, 6143, 6163, 6199, 6203, 6211, 6247, 6263, 6271, 6287, 6299, 6323, 6343, 6359, 6367, 6379, 6451, 6491, 6547, 6551, 6599, 6607, 6619, 6659, 6679, 6691, 6703, 6719, 6763, 6803, 6823, 6827, 6863, 6871, 6883, 6899, 6907, 6911, 6947, 6959, 6967, 6971, 6983, 6991, 7027, 7039, 7043, 7079, 7103, 7127, 7151, 7159, 7187, 7207, 7211, 7219, 7243, 7247, 7283, 7307, 7351, 7411, 7451, 7487, 7499, 7507, 7523, 7547, 7559, 7583, 7603, 7607, 7687, 7691, 7699, 7703, 7723, 7727, 7759, 7823, 7867, 7907, 7927, 7951, 7963, 8011, 8039, 8087, 8111, 8123, 8147, 8167, 8171, 8179, 8191, 8219, 8231, 8243, 8263, 8287, 8291, 8387, 8419, 8423, 8431, 8443, 8467, 8527, 8563, 8599, 8627, 8663, 8699, 8719, 8747, 8779, 8783, 8807, 8819, 8831, 8839, 8867, 8887, 8923, 8951, 8963, 8971, 9007, 9011, 9043, 9059, 9067, 9091, 9103, 9127, 9187, 9199, 9203, 9227, 9239, 9283, 9311, 9319, 9343, 9391, 9403, 9439, 9463, 9491, 9511, 9547, 9587, 9619, 9623, 9631, 9643, 9679, 9719, 9739, 9743, 9767, 9787, 9791, 9811, 9839, 9851, 9859, 9883, 9887, 9923, 9931, 9967, 10007, 10079, 10091, 10103, 10111, 10139, 10151, 10159, 10163, 10211, 10223, 10247, 10259, 10267, 10303, 10331, 10343, 10391, 10399, 10427, 10459, 10463, 10487, 10499, 10531, 10559, 10627, 10631, 10639, 10651, 10663, 10667, 10687, 10691, 10711, 10723, 10771, 10799, 10831, 10859, 10867, 10883, 10891, 10903, 10979, 10987, 11003, 11047, 11059, 11071, 11083, 11087, 11131, 11159, 11171, 11239, 11243, 11251, 11279, 11287, 11299, 11351, 11383, 11399, 11411, 11423, 11443, 11483, 11503, 11519, 11527, 11551, 11579, 11587, 11699, 11719, 11731, 11743, 11779, 11783, 11807, 11827, 11831, 11839, 11863, 11867, 11887, 11903, 11923, 11927, 11939, 11959, 11971, 11987, 12011, 12043, 12107, 12119, 12143, 12163, 12211, 12343, 12347, 12379, 12451, 12479, 12487, 12491, 12503, 12511, 12527, 12539, 12547, 12583, 12611, 12647, 12659, 12671, 12703, 12739, 12743, 12763, 12791, 12799, 12823, 12899, 12907, 12911, 12919, 12923, 12959, 12967, 12979, 12983, 13007, 13063, 13103, 13127, 13147, 13151, 13159, 13163, 13183, 13187, 13219, 13259, 13267, 13291, 13327, 13331, 13367, 13399, 13411, 13451, 13463, 13487, 13523, 13567, 13591, 13619, 13627, 13679, 13711, 13751, 13759, 13807, 13831, 13859, 13879, 13883, 13903, 13907, 13931, 13963, 13967, 13999, 14011, 14051, 14071, 14083, 14087, 14143, 14207, 14243, 14251, 14303, 14323, 14327, 14347, 14407, 14411, 14419, 14423, 14431, 14447, 14479, 14519, 14543, 14551, 14563, 14591, 14627, 14683, 14699, 14723, 14731, 14747, 14759, 14767, 14771, 14779, 14843, 14851, 14887, 14891, 14923, 14939, 14951, 14983, 15031, 15091, 15107, 15139, 15187, 15199, 15259, 15263, 15271, 15287, 15299, 15319, 15331, 15359, 15383, 15391, 15427, 15439, 15443, 15467, 15527, 15551, 15559, 15583, 15607, 15647, 15667, 15671, 15679, 15683, 15727, 15731, 15739, 15767, 15787, 15791, 15803, 15823, 15907, 15919, 15923, 15959, 15971, 15991, 16007, 16063, 16091, 16103, 16139, 16183, 16187, 16223, 16267, 16319, 16363, 16411, 16427, 16447, 16487, 16519, 16547, 16567, 16603, 16607, 16619, 16631, 16651, 16691, 16699, 16703, 16747, 16759, 16763, 16787, 16811, 16823, 16831, 16843, 16871, 16879, 16883, 16903, 16927, 16931, 16943, 16963, 16979, 17011, 17027, 17047, 17099, 17107, 17123, 17167, 17183, 17191, 17203, 17207, 17231, 17239, 17291, 17327, 17351, 17359, 17383, 17387, 17419, 17431, 17443, 17467, 17471, 17483, 17491, 17519, 17539, 17551, 17579, 17599, 17623, 17627, 17659, 17683, 17707, 17747, 17807, 17827, 17839, 17851, 17863, 17891, 17903, 17911, 17923, 17939, 17959, 17971, 17987, 18043, 18047, 18059, 18119, 18127, 18131, 18143, 18199, 18211, 18251, 18307, 18311, 18367, 18371, 18379, 18427, 18443, 18451, 18503, 18539, 18583, 18671, 18691, 18719, 18731, 18743, 18787, 18839, 18859, 18899, 18919, 18947, 18959, 18979, 19031, 19051, 19079, 19139, 19163, 19183, 19207, 19211, 19219, 19231, 19259, 19267, 19379, 19387, 19391, 19403, 19423, 19427, 19447, 19463, 19471, 19483, 19531, 19559, 19571, 19583, 19603, 19687, 19699, 19727, 19739, 19751, 19763, 19819, 19843, 19867, 19927, 19963, 19979, 19991, 20011, 20023, 20047, 20051, 20063, 20071, 20107, 20143, 20147, 20183, 20219, 20231, 20327, 20347, 20359, 20399, 20411, 20431, 20443, 20479, 20483, 20543, 20551, 20563, 20599, 20611, 20627, 20639, 20663, 20719, 20743, 20759, 20771, 20807, 20879, 20899, 20903, 20947, 20959, 20963, 20983, 21011, 21019, 21059, 21107, 21187, 21211, 21227, 21247, 21319, 21323, 21379, 21383, 21391, 21419, 21467, 21491, 21499, 21523, 21559, 21563, 21587, 21599, 21647, 21683, 21727, 21739, 21767, 21787, 21799, 21803, 21839, 21851, 21859, 21863, 21871, 21911, 21943, 21991, 22027, 22039, 22051, 22063, 22067, 22091, 22111, 22123, 22159, 22171, 22259, 22271, 22279, 22291, 22303, 22307, 22367, 22391, 22447, 22483, 22511, 22531, 22543, 22567, 22571, 22639, 22643, 22651, 22679, 22691, 22699, 22727, 22739, 22751, 22783, 22787, 22807, 22811, 22859, 22943, 22963, 23027, 23039, 23063, 23071, 23087, 23099, 23131, 23143, 23159, 23167, 23203, 23227, 23251, 23279, 23291, 23311, 23327, 23339, 23371, 23431, 23459, 23563, 23567, 23599, 23603, 23623, 23663, 23671, 23687, 23719, 23747, 23827, 23879, 23887, 23899, 23971, 24007, 24019, 24043, 24071, 24083, 24103, 24107, 24151, 24179, 24203, 24223, 24239, 24247, 24371, 24379, 24391, 24407, 24419, 24439, 24443, 24527, 24547, 24571, 24611, 24631, 24671, 24683, 24763, 24767, 24799, 24851, 24907, 24919, 24923, 24943, 24967, 24971, 24979, 25087, 25111, 25127, 25147, 25163, 25171, 25183, 25219, 25243, 25247, 25307, 25339, 25343, 25367, 25423, 25439, 25447, 25471, 25523, 25579, 25583, 25603, 25639, 25643, 25667, 25703, 25747, 25759, 25763, 25771, 25799, 25819, 25847, 25867, 25931, 25939, 25951, 25999, 26003, 26099, 26111, 26171, 26203, 26227, 26263, 26267, 26339, 26347, 26371, 26399, 26431, 26459, 26479, 26539, 26591, 26627, 26647, 26699, 26711, 26723, 26731, 26759, 26783, 26839, 26863, 26879, 26891, 26903, 26927, 26947, 26951, 26959, 26987, 27011, 27031, 27043, 27059, 27067, 27107, 27127, 27143, 27179, 27191, 27259, 27271, 27283, 27299, 27367, 27407, 27427, 27431, 27487, 27527, 27539, 27551, 27611, 27631, 27647, 27691, 27739, 27743, 27751, 27763, 27767, 27779, 27791, 27799, 27803, 27823, 27827, 27847, 27851, 27883, 27919, 27947, 27967, 28019, 28027, 28031, 28087, 28099, 28111, 28123, 28163, 28211, 28279, 28283, 28307, 28319, 28351, 28387, 28403, 28439, 28447, 28463, 28499, 28547, 28571, 28579, 28591, 28603, 28619, 28627, 28631, 28663, 28687, 28703, 28711, 28723, 28759, 28771, 28807, 28843, 28867, 28871, 28879, 28979, 29023, 29027, 29059, 29147, 29167, 29179, 29231, 29251, 29287, 29303, 29311, 29327, 29339, 29347, 29363, 29383, 29387, 29399, 29411, 29423, 29443, 29483, 29527, 29531, 29567, 29587, 29599, 29663, 29723, 29759, 29803, 29819, 29863, 29867, 29879, 29947, 29959, 29983, 30011, 30047, 30059, 30071, 30091, 30103, 30119, 30139, 30187, 30203, 30211, 30223, 30259, 30307, 30319, 30323, 30347, 30391, 30403, 30427, 30431, 30491, 30539, 30631, 30643, 30671, 30703, 30707, 30727, 30839, 30871, 30911, 30971, 30983, 31051, 31063, 31091, 31123, 31147, 31151, 31159, 31183, 31219, 31231, 31247, 31267, 31271, 31379, 31387, 31511, 31531, 31543, 31547, 31567, 31583, 31627, 31643, 31663, 31667, 31687, 31699, 31723, 31727, 31751, 31799, 31859, 31883, 31891, 31907, 31963, 31991, 32051, 32059, 32063, 32083, 32099, 32119, 32143, 32159, 32191, 32203, 32251, 32303, 32323, 32327, 32363, 32371, 32411, 32423, 32443, 32479, 32491, 32507, 32531, 32563, 32587, 32603, 32611, 32647, 32687, 32719, 32779, 32783, 32803, 32831, 32839, 32843, 32887, 32911, 32971, 32983, 32987, 32999, 33023, 33071, 33083, 33091, 33107, 33119, 33191, 33199, 33203, 33211, 33223, 33247, 33311, 33331, 33343, 33347, 33359, 33391, 33403, 33427, 33479, 33503, 33547, 33563, 33587, 33599, 33619, 33623, 33647, 33679, 33703, 33751, 33767, 33791, 33827, 33863, 33871, 33911, 33923, 33931, 33967, 34019, 34031, 34039, 34123, 34159, 34171, 34183, 34211, 34231, 34267, 34283, 34303, 34319, 34327, 34351, 34403, 34439, 34487, 34499, 34511, 34519, 34543, 34583, 34591, 34603, 34607, 34631, 34651, 34679, 34703, 34739, 34747, 34759, 34763, 34807, 34819, 34843, 34847, 34871, 34919, 34963, 35027, 35051, 35083, 35099, 35111, 35159, 35171, 35227, 35251, 35267, 35291, 35311, 35323, 35327, 35339, 35363, 35407, 35419, 35423, 35447, 35507, 35527, 35531, 35543, 35591, 35603, 35671, 35731, 35747, 35771, 35803, 35831, 35851, 35863, 35899, 35911, 35923, 35951, 35983, 35999, 36007, 36067, 36083, 36107, 36131, 36151, 36187, 36191, 36251, 36299, 36319, 36343, 36383, 36451, 36467, 36523, 36527, 36563, 36571, 36583, 36587, 36599, 36607, 36643, 36671, 36683, 36691, 36767, 36787, 36847, 36887, 36899, 36919, 36923, 36931, 36943, 36979, 37003, 37019, 37039, 37087, 37123, 37139, 37159, 37171, 37199, 37223, 37243, 37307, 37339, 37363, 37379, 37447, 37463, 37483, 37511, 37547, 37567, 37571, 37579, 37591, 37607, 37619, 37643, 37663, 37691, 37699, 37747, 37783, 37799, 37831, 37847, 37871, 37879, 37907, 37951, 37963, 37967, 37987, 37991, 38011, 38039, 38047, 38083, 38119, 38167, 38183, 38219, 38231, 38239, 38287, 38299, 38303, 38327, 38351, 38431, 38447, 38459, 38543, 38567, 38603, 38611, 38639, 38651, 38671, 38699, 38707, 38723, 38747, 38767, 38783, 38803, 38839, 38851, 38867, 38891, 38903, 38923, 38959, 39019, 39023, 39043, 39047, 39103, 39107, 39119, 39163, 39191, 39199, 39227, 39239, 39251, 39323, 39343, 39359, 39367, 39383, 39439, 39443, 39451, 39499, 39503, 39511, 39551, 39563, 39607, 39619, 39623, 39631, 39659, 39667, 39671, 39679, 39703, 39719, 39727, 39779, 39791, 39799, 39827, 39839, 39863, 39883, 39887, 39971, 39979, 40031, 40039, 40063, 40099, 40123, 40151, 40163, 40231, 40283, 40351, 40387, 40423, 40427, 40459, 40471, 40483, 40487, 40499, 40559, 40583, 40591, 40639, 40699, 40739, 40751, 40759, 40763, 40771, 40787, 40823, 40847, 40879, 40883, 40903, 40927, 41011, 41023, 41051, 41131, 41143, 41179, 41183, 41203, 41227, 41231, 41243, 41299, 41387, 41399, 41411, 41443, 41467, 41479, 41491, 41519, 41543, 41579, 41603, 41611, 41627, 41647, 41651, 41659, 41687, 41771, 41863, 41887, 41911, 41927, 41947, 41983, 41999, 42019, 42043, 42131, 42139, 42179, 42223, 42227, 42239, 42283, 42307, 42323, 42331, 42359, 42379, 42391, 42403, 42407, 42451, 42467, 42491, 42499, 42571, 42643, 42667, 42719, 42743, 42751, 42839, 42859, 42863, 42899, 42923, 42943, 42967, 43003, 43051, 43067, 43103, 43151, 43159, 43207, 43223, 43271, 43283, 43331, 43391, 43399, 43411, 43487, 43543, 43579, 43607, 43691, 43759, 43783, 43943, 43963, 43987, 44027, 44059, 44071, 44087, 44111, 44119, 44123, 44131, 44159, 44171, 44179, 44203, 44207, 44263, 44267, 44279, 44351, 44371, 44383, 44483, 44491, 44507, 44543, 44579, 44587, 44623, 44647, 44651, 44687, 44699, 44711, 44819, 44839, 44843, 44851, 44867, 44879, 44887, 44927, 44959, 44963, 44971, 45007, 45083, 45119, 45127, 45131, 45247, 45259, 45263, 45307, 45319, 45343, 45403, 45439, 45491, 45523, 45587, 45599, 45631, 45667, 45691, 45707, 45751, 45763, 45767, 45779, 45823, 45827, 45887, 45943, 45959, 45971, 46027, 46051, 46091, 46099, 46103, 46147, 46171, 46183, 46187, 46199, 46219, 46271, 46279, 46327, 46351, 46399, 46411, 46439, 46451, 46499, 46507, 46511, 46523, 46567, 46619, 46639, 46643, 46679, 46687, 46703, 46723, 46727, 46747, 46751, 46771, 46807, 46811, 46819, 46831, 46867, 46919, 47051, 47059, 47123, 47147, 47207, 47251, 47303, 47339, 47363, 47387, 47407, 47419, 47431, 47459, 47491, 47507, 47527, 47543, 47591, 47599, 47623, 47639, 47659, 47699, 47743, 47779, 47791, 47807, 47843, 47903, 47911, 47947, 48023, 48079, 48091, 48119, 48131, 48163, 48179, 48187, 48239, 48259, 48271, 48299, 48311, 48371, 48383, 48407, 48463, 48487, 48491, 48523, 48527, 48571, 48611, 48619, 48647, 48731, 48767, 48779, 48799, 48823, 48847, 48859, 48871, 48907, 48991, 49003, 49019, 49043, 49103, 49123, 49139, 49171, 49199, 49211, 49223, 49279, 49307, 49331, 49339, 49363, 49391, 49451, 49459, 49463, 49499, 49523, 49547, 49559, 49603, 49639, 49663, 49667, 49739, 49747, 49783, 49787, 49807, 49811, 49823, 49831, 49843, 49871, 49891, 49919, 49927, 49943, 49991, 49999, 50023, 50047, 50051, 50087, 50111, 50131, 50159, 50207, 50227, 50231, 50263, 50287, 50291, 50311, 50359, 50363, 50383, 50387, 50423, 50503, 50527, 50539, 50543, 50551, 50587, 50591, 50599, 50647, 50651, 50683, 50707, 50767, 50839, 50867, 50891, 50951, 50971, 51031, 51043, 51047, 51059, 51071, 51131, 51151, 51199, 51203, 51239, 51263, 51283, 51287, 51343, 51347, 51407, 51427, 51439, 51479, 51487, 51503, 51511, 51539, 51599, 51607, 51631, 51647, 51659, 51679, 51719, 51767, 51787, 51839, 51871, 51907, 51971, 51991, 52027, 52051, 52067, 52127, 52147, 52163, 52223, 52259, 52267, 52291, 52363, 52379, 52387, 52391, 52511, 52543, 52567, 52571, 52579, 52583, 52631, 52639, 52667, 52691, 52711, 52883, 52903, 52919, 52951, 52963, 52967, 52999, 53047, 53051, 53087, 53147, 53171, 53231, 53239, 53279, 53299, 53323, 53327, 53411, 53419, 53479, 53503, 53527, 53551, 53591, 53611, 53623, 53639, 53699, 53719, 53731, 53759, 53783, 53791, 53819, 53831, 53887, 53891, 53899, 53923, 53951, 53959, 53987, 54011, 54083, 54091, 54139, 54163, 54167, 54251, 54311, 54323, 54331, 54367, 54419, 54443, 54499, 54503, 54547, 54559, 54563, 54583, 54623, 54631, 54647, 54667, 54679, 54727, 54751, 54767, 54779, 54787, 54851, 54907, 54979, 55051, 55079, 55103, 55127, 55147, 55171, 55219, 55243, 55259, 55291, 55331, 55343, 55351, 55399, 55411, 55439, 55487, 55511, 55547, 55579, 55619, 55631, 55639, 55691, 55799, 55807, 55819, 55823, 55843, 55931, 55967, 55987, 56003, 56087, 56099, 56123, 56131, 56179, 56207, 56239, 56263, 56311, 56359, 56431, 56443, 56467, 56519, 56527, 56531, 56543, 56591, 56611, 56659, 56663, 56671, 56711, 56731, 56747, 56767, 56779, 56783, 56807, 56827, 56843, 56891, 56911, 56923, 56951, 56963, 56999, 57047, 57059, 57107, 57131, 57139, 57143, 57163, 57179, 57191, 57203, 57223, 57251, 57259, 57287, 57347, 57367, 57383, 57427, 57467, 57487, 57503, 57559, 57571, 57587, 57667, 57679, 57719, 57751, 57791, 57803, 57839, 57859, 57899, 57923, 57943, 57947, 57991, 58027, 58043, 58067, 58111, 58171, 58199, 58207, 58211, 58243, 58363, 58367, 58379, 58411, 58427, 58439, 58451, 58511, 58543, 58679, 58687, 58699, 58711, 58727, 58763, 58771, 58787, 58943, 58963, 58967, 58991, 59011, 59063, 59083, 59107, 59119, 59123, 59159, 59167, 59183, 59207, 59219, 59239, 59263, 59359, 59387, 59447, 59467, 59471, 59567, 59627, 59651, 59659, 59663, 59671, 59707, 59723, 59771, 59779, 59791, 59863, 59879, 59887, 59951, 59971, 59999, 60083, 60091, 60107, 60127, 60139, 60167, 60223, 60251, 60271, 60331, 60343, 60383, 60427, 60443, 60527, 60539, 60607, 60611, 60631, 60647, 60659, 60679, 60703, 60719, 60763, 60779, 60811, 60859, 60887, 60899, 60919, 60943, 61027, 61031, 61051, 61099, 61151, 61223, 61231, 61291, 61331, 61343, 61363, 61379, 61403, 61463, 61471, 61483, 61487, 61507, 61511, 61519, 61543, 61559, 61583, 61603, 61627, 61643, 61651, 61667, 61687, 61703, 61819, 61843, 61871, 61879, 61927, 61967, 61987, 61991, 62011, 62039, 62047, 62099, 62119, 62131, 62143, 62171, 62207, 62299, 62303, 62311, 62323, 62347, 62351, 62383, 62459, 62483, 62507, 62563, 62603, 62627, 62639, 62659, 62683, 62723, 62731, 62791, 62819, 62827, 62851, 62927, 62939, 62971, 62983, 62987, 63031, 63059, 63067, 63079, 63103, 63127, 63131, 63179, 63199, 63211, 63247, 63299, 63311, 63331, 63347, 63367, 63391, 63419, 63439, 63443, 63463, 63467, 63487, 63499, 63559, 63587, 63599, 63611, 63647, 63659, 63667, 63671, 63691, 63703, 63719, 63743, 63799, 63803, 63823, 63839, 63863, 64019, 64063, 64067, 64091, 64123, 64151, 64171, 64187, 64223, 64271, 64279, 64283, 64303, 64319, 64327, 64399, 64403, 64439, 64483, 64591, 64627, 64663, 64667, 64679, 64747, 64763, 64783, 64811, 64871, 64919, 64927, 65011, 65063, 65071, 65099, 65111, 65119, 65123, 65147, 65167, 65171, 65179, 65203, 65239, 65267, 65323, 65327, 65371, 65407, 65423, 65447, 65479, 65519, 65539, 65543, 65551, 65579, 65587, 65599, 65647, 65651, 65687, 65699, 65719, 65731, 65827, 65831, 65839, 65843, 65851, 65867, 65951, 65963, 65983, 66071, 66083, 66107, 66179, 66191, 66271, 66347, 66359, 66383, 66431, 66467, 66491, 66499, 66571, 66643, 66683, 66739, 66751, 66763, 66863, 66919, 66923, 66931, 66943, 66947, 66959, 67043, 67139, 67187, 67211, 67219, 67247, 67271, 67307, 67339, 67343, 67391, 67399, 67411, 67447, 67499, 67511, 67523, 67531, 67559, 67567, 67579, 67619, 67631, 67651, 67679, 67699, 67723, 67751, 67763, 67783, 67807, 67819, 67843, 67883, 67891, 67927, 67931, 67939, 67967, 67987, 68023, 68059, 68071, 68087, 68099, 68111, 68171, 68207, 68227, 68239, 68311, 68351, 68371, 68399, 68447, 68491, 68507, 68543, 68567, 68611, 68639, 68659, 68683, 68687, 68699, 68711, 68743, 68771, 68863, 68879, 68891, 68927, 68963, 69011, 69019, 69031, 69067, 69127, 69143, 69151, 69163, 69191, 69203, 69247, 69259, 69263, 69371, 69379, 69427, 69431, 69439, 69463, 69467, 69499, 69539, 69623, 69691, 69739, 69763, 69779, 69827, 69847, 69859, 69911, 69931, 69959, 69991, 70003, 70051, 70067, 70079, 70099, 70111, 70139, 70163, 70199, 70207, 70271, 70327, 70351, 70423, 70439, 70451, 70459, 70487, 70507, 70571, 70583, 70607, 70627, 70663, 70667, 70687, 70823, 70843, 70867, 70891, 70919, 70951, 70991, 70999, 71011, 71023, 71039, 71119, 71143, 71167, 71171, 71191, 71263, 71339, 71347, 71359, 71363, 71387, 71399, 71411, 71419, 71443, 71471, 71479, 71483, 71503, 71527, 71551, 71671, 71699, 71707, 71719, 71807, 71843, 71867, 71887, 71899, 71947, 71963, 71971, 71983, 71999, 72019, 72031, 72043, 72047, 72167, 72211, 72223, 72227, 72251, 72271, 72287, 72307, 72367, 72379, 72383, 72431, 72467, 72503, 72547, 72559, 72623, 72643, 72647, 72671, 72679, 72707, 72719, 72727, 72739, 72763, 72767, 72823, 72859, 72871, 72883, 72907, 72911, 72923, 72931, 72959, 73019, 73039, 73043, 73063, 73091, 73259, 73291, 73303, 73327, 73351, 73363, 73379, 73387, 73459, 73471, 73483, 73523, 73571, 73583, 73607, 73643, 73651, 73699, 73727, 73771, 73783, 73823, 73847, 73859, 73867, 73883, 73907, 73939, 73943, 73951, 73999, 74027, 74047, 74071, 74099, 74131, 74143, 74159, 74219, 74231, 74279, 74287, 74311, 74323, 74363, 74383, 74419, 74471, 74507, 74551, 74587, 74611, 74623, 74687, 74699, 74707, 74719, 74747, 74759, 74771, 74827, 74843, 74887, 74891, 74903, 74923, 74959, 75011, 75079, 75083, 75167, 75211, 75223, 75227, 75239, 75307, 75323, 75367, 75391, 75403, 75407, 75503, 75511, 75527, 75539, 75583, 75611, 75659, 75703, 75707, 75731, 75787, 75967, 75979, 75983, 75991, 76031, 76039, 76079, 76091, 76103, 76123, 76147, 76159, 76163, 76207, 76243, 76259, 76303, 76343, 76367, 76379, 76403, 76423, 76463, 76471, 76507, 76511, 76519, 76543, 76579, 76603, 76607, 76631, 76651, 76667, 76679, 76771, 76819, 76847, 76871, 76907, 76919, 76943, 76963, 77023, 77047, 77167, 77171, 77191, 77239, 77243, 77267, 77279, 77291, 77323, 77347, 77351, 77383, 77419, 77431, 77447, 77471, 77527, 77551, 77563, 77587, 77647, 77659, 77699, 77711, 77719, 77731, 77743, 77747, 77783, 77863, 77867, 77899, 77951, 77983, 77999, 78007, 78031, 78079, 78139, 78163, 78179, 78191, 78203, 78283, 78307, 78427, 78439, 78479, 78487, 78511, 78539, 78583, 78607, 78643, 78779, 78791, 78803, 78823, 78839, 78887, 78979, 79031, 79039, 79043, 79063, 79087, 79103, 79139, 79147, 79151, 79159, 79231, 79283, 79319, 79367, 79411, 79423, 79427, 79451, 79531, 79559, 79579, 79627, 79631, 79687, 79691, 79699, 79811, 79823, 79843, 79847, 79903, 79939, 79967, 79979, 79987, 79999, 80039, 80051, 80071, 80111, 80167, 80191, 80207, 80231, 80239, 80251, 80263, 80287, 80347, 80363, 80387, 80407, 80447, 80471, 80527, 80567, 80599, 80603, 80611, 80627, 80651, 80671, 80683, 80687, 80747, 80783, 80819, 80831, 80863, 80911, 80923, 80963, 81019, 81023, 81031, 81071, 81083, 81119, 81131, 81199, 81203, 81239, 81283, 81299, 81307, 81331, 81343, 81359, 81371, 81439, 81463, 81527, 81547, 81563, 81611, 81619, 81647, 81667, 81707, 81727, 81799, 81839, 81847, 81883, 81899, 81919, 81943, 81967, 81971, 82003, 82007, 82031, 82039, 82051, 82067, 82163, 82171, 82183, 82207, 82219, 82223, 82231, 82267, 82279, 82307, 82339, 82351, 82387, 82463, 82471, 82487, 82499, 82507, 82531, 82559, 82567, 82571, 82591, 82619, 82651, 82699, 82723, 82727, 82763, 82799, 82811, 82847, 82883, 82891, 82903, 82939, 82963, 83023, 83047, 83059, 83063, 83071, 83203, 83219, 83227, 83231, 83267, 83311, 83339, 83383, 83407, 83431, 83459, 83471, 83579, 83591, 83719, 83791, 83891, 83903, 83939, 83987, 84011, 84047, 84059, 84067, 84127, 84131, 84163, 84179, 84191, 84199, 84211, 84223, 84239, 84247, 84299, 84307, 84347, 84443, 84463, 84467, 84499, 84503, 84551, 84559, 84659, 84691, 84719, 84731, 84751, 84787, 84811, 84859, 84947, 84967, 84991, 85027, 85087, 85091, 85103, 85147, 85159, 85199, 85223, 85247, 85303, 85331, 85363, 85439, 85447, 85487, 85523, 85531, 85607, 85639, 85643, 85667, 85691, 85703, 85711, 85751, 85819, 85831, 85843, 85931, 85991, 85999, 86011, 86027, 86083, 86131, 86143, 86171, 86179, 86239, 86243, 86263, 86311, 86323, 86351, 86371, 86399, 86423, 86467, 86491, 86531, 86579, 86587, 86599, 86711, 86719, 86743, 86767, 86783, 86843, 86851, 86923, 86927, 86939, 86951, 86959, 87011, 87071, 87103, 87119, 87151, 87187, 87211, 87223, 87299, 87323, 87359, 87383, 87403, 87427, 87443, 87491, 87523, 87539, 87547, 87559, 87583, 87623, 87631, 87671, 87679, 87683, 87719, 87739, 87743, 87751, 87803, 87811, 87887, 87931, 87991, 88007, 88019, 88079, 88259, 88327, 88411, 88423, 88463, 88471, 88499, 88523, 88591, 88607, 88643, 88663, 88667, 88771, 88799, 88819, 88843, 88883, 88903, 88919, 88951, 89071, 89083, 89087, 89107, 89119, 89123, 89203, 89227, 89231, 89303, 89363, 89371, 89387, 89431, 89443, 89459, 89519, 89527, 89567, 89591, 89603, 89611, 89627, 89659, 89671, 89767, 89779, 89783, 89819, 89867, 89891, 89899, 89923, 89939, 89959, 89963, 89983, 90007, 90011, 90019, 90023, 90031, 90059, 90067, 90071, 90107, 90127, 90163, 90187, 90191, 90199, 90203, 90227, 90239, 90247, 90359, 90371, 90379, 90403, 90407, 90499, 90511, 90523, 90547, 90583, 90619, 90631, 90647, 90659, 90679, 90703, 90731, 90787, 90803, 90823, 90847, 90863, 90887, 90907, 90911, 90931, 90947, 90971, 91019, 91099, 91127, 91139, 91151, 91159, 91163, 91183, 91199, 91243, 91283, 91303, 91367, 91387, 91411, 91423, 91459, 91499, 91571, 91583, 91591, 91631, 91639, 91691, 91703, 91711, 91771, 91823, 91867, 91939, 91943, 92051, 92083, 92107, 92111, 92119, 92143, 92203, 92227, 92243, 92251, 92311, 92347, 92363, 92383, 92399, 92419, 92431, 92459, 92467, 92479, 92507, 92551, 92567, 92623, 92627, 92639, 92647, 92671, 92683, 92699, 92707, 92723, 92767, 92791, 92831, 92863, 92867, 92927, 92951, 92987, 93047, 93059, 93083, 93103, 93131, 93139, 93151, 93179, 93187, 93199, 93239, 93251, 93307, 93319, 93323, 93371, 93383, 93407, 93419, 93427, 93463, 93479, 93487, 93491, 93523, 93559, 93563, 93607, 93683, 93703, 93719, 93763, 93787, 93811, 93827, 93851, 93871, 93887, 93911, 93971, 93983, 94007, 94063, 94079, 94099, 94111, 94151, 94207, 94219, 94291, 94307, 94331, 94343, 94351, 94399, 94427, 94439, 94447, 94483, 94531, 94543, 94583, 94603, 94687, 94723, 94727, 94747, 94771, 94811, 94819, 94823, 94847, 94903, 94907, 94951, 94999, 95003, 95063, 95071, 95083, 95087, 95107, 95111, 95131, 95143, 95191, 95203, 95239, 95267, 95279, 95287, 95311, 95339, 95383, 95419, 95443, 95507, 95527, 95531, 95539, 95603, 95651, 95747, 95783, 95791, 95803, 95819, 95891, 95911, 95923, 95947, 95959, 95971, 96043, 96059, 96079, 96167, 96179, 96199, 96223, 96259, 96263, 96323, 96331, 96419, 96431, 96451, 96479, 96487, 96527, 96587, 96643, 96667, 96671, 96703, 96739, 96779, 96787, 96799, 96823, 96827, 96847, 96851, 96907, 96931, 96959, 96979, 97007, 97151, 97159, 97171, 97187, 97231, 97259, 97283, 97303, 97327, 97367, 97379, 97387, 97423, 97459, 97463, 97499, 97511, 97523, 97547, 97571, 97579, 97583, 97651, 97687, 97711, 97771, 97787, 97843, 97859, 97871, 97879, 97883, 97919, 97927, 97931, 97943, 97987, 98011, 98123, 98179, 98207, 98227, 98323, 98347, 98407, 98411, 98419, 98459, 98467, 98479, 98491, 98519, 98543, 98563, 98627, 98663, 98711, 98731, 98779, 98807, 98867, 98899, 98911, 98927, 98939, 98947, 98963, 98999, 99023, 99079, 99103, 99131, 99139, 99251, 99259, 99347, 99367, 99431, 99523, 99527, 99551, 99559, 99563, 99571, 99607, 99611, 99623, 99643, 99667, 99679, 99707, 99767, 99787, 99823, 99839, 99859, 99871, 99907, 99923, 99971, 99991
類数 3 (508)
$\operatorname{Cl}(K) \cong C_3$ (508)
79, 223, 359, 443, 659, 839, 1091, 1171, 1223, 1367, 1523, 1567, 1627, 1787, 1811, 1847, 1907, 1987, 2099, 2143, 2207, 2459, 2543, 2659, 2711, 2971, 3023, 3163, 3391, 3739, 3803, 4159, 4283, 4651, 5011, 5099, 5303, 5431, 5503, 5683, 5903, 6311, 6427, 6571, 6791, 7019, 7459, 7639, 7643, 7879, 7883, 8363, 8447, 8543, 8623, 8731, 8863, 8999, 9151, 9323, 9419, 9467, 9479, 9551, 9907, 10039, 10099, 10271, 10939, 11119, 11311, 11467, 11471, 12007, 12071, 12203, 12227, 12239, 12263, 13043, 13099, 13339, 13723, 13799, 14107, 14387, 14783, 14867, 14879, 15083, 15227, 15451, 15511, 15643, 15859, 16067, 16087, 16231, 17299, 17791, 18287, 18523, 18911, 19759, 20123, 20287, 20323, 20507, 20707, 20731, 20747, 20887, 21031, 21067, 21139, 21143, 21283, 21347, 21487, 21503, 21751, 22079, 22147, 22247, 22343, 22619, 22907, 23011, 23399, 23447, 23531, 23627, 23743, 23767, 23831, 23911, 24091, 24499, 24551, 24623, 24659, 24691, 24847, 25031, 25303, 25411, 25679, 25903, 25943, 26083, 26107, 26119, 26183, 26407, 26423, 26683, 26687, 27091, 27103, 27583, 27943, 27983, 28051, 28219, 28607, 28927, 29063, 29123, 29131, 29191, 29207, 29611, 29671, 30271, 30467, 30559, 30763, 30803, 30851, 30931, 31019, 31039, 31079, 31139, 31223, 31259, 31307, 31319, 31607, 32003, 32027, 32467, 32503, 32579, 32707, 32771, 32939, 33151, 33287, 33811, 33851, 34127, 34147, 34259, 34667, 34687, 34939, 35107, 35879, 35963, 36011, 36307, 36871, 36947, 37811, 38791, 38971, 39079, 39139, 39371, 39419, 39847, 39983, 40111, 40127, 40519, 41047, 41263, 41507, 41719, 41903, 41959, 42187, 42443, 42463, 42487, 42611, 42683, 42767, 42979, 43019, 43291, 43427, 43627, 43651, 43711, 43891, 44531, 44683, 44771, 44939, 44987, 45139, 45503, 45659, 45979, 46307, 46447, 46591, 46691, 47119, 47143, 47279, 47351, 47939, 47951, 48479, 48623, 48751, 48787, 48947, 49031, 49411, 49627, 50119, 50123, 50411, 50459, 50671, 50723, 51383, 51551, 51691, 51803, 51827, 51859, 52183, 52727, 52807, 52859, 53003, 53267, 53927, 54059, 54151, 54319, 54347, 54403, 54539, 54799, 54919, 54959, 54983, 55207, 55663, 55763, 55871, 56039, 56267, 56479, 56503, 56599, 56687, 56983, 57283, 57331, 57527, 57847, 58151, 58579, 58831, 58907, 59243, 59351, 59419, 59443, 61043, 61211, 61283, 61547, 62219, 62467, 62591, 62903, 63527, 63607, 63727, 63907, 64579, 64879, 66067, 66103, 66463, 66523, 66791, 67103, 67231, 67547, 67607, 67759, 67867, 67943, 67979, 68443, 68483, 68531, 68767, 69119, 69239, 69383, 69767, 70039, 70123, 70379, 70783, 70879, 70979, 71147, 71327, 71563, 71647, 71663, 72091, 72103, 73079, 73127, 73243, 73679, 73819, 74051, 74167, 74203, 74411, 74567, 74779, 75347, 75479, 75571, 75619, 75679, 75683, 75743, 75883, 75931, 76003, 76099, 76487, 76831, 76883, 76991, 77003, 77339, 77359, 77479, 77591, 77687, 77723, 78059, 78167, 78311, 78367, 78623, 78707, 78919, 79111, 79187, 79399, 79867, 79907, 80107, 80147, 80491, 81043, 81047, 81163, 81671, 81931, 82483, 83003, 83299, 83399, 83423, 83443, 83563, 83983, 84143, 84407, 84431, 84631, 84827, 84919, 85243, 85259, 85571, 85619, 85903, 86111, 86183, 86291, 86771, 87083, 87107, 87179, 87251, 87691, 87767, 87911, 87943, 87959, 88003, 88379, 88547, 88747, 88807, 89563, 89759, 89839, 90263, 90527, 91079, 91291, 91331, 91951, 91967, 92003, 92179, 92219, 92387, 92503, 92779, 92899, 92959, 93263, 93287, 93979, 94379, 94559, 94651, 95327, 95467, 95707, 95731, 95987, 96443, 96911, 97039, 97103, 97847, 98047, 98251, 98443, 98507, 98639, 99119, 99191, 99371, 99391, 99487
類数 5 (180)
$\operatorname{Cl}(K) \cong C_5$ (180)
439, 499, 727, 1327, 2027, 2399, 3251, 3967, 4271, 4591, 5107, 5527, 5711, 5927, 6563, 6779, 7331, 7591, 8539, 8707, 8803, 9371, 9431, 9803, 10067, 10243, 10567, 10739, 10847, 12251, 13171, 13499, 13763, 14831, 15307, 15619, 15887, 16111, 16451, 17783, 18191, 18439, 18679, 18803, 19543, 19891, 19919, 20407, 20939, 21163, 21179, 21407, 22283, 22871, 23059, 25391, 26387, 27239, 27479, 28183, 28643, 28859, 31847, 32183, 33739, 34367, 34471, 34483, 35279, 35491, 35839, 36791, 37423, 40867, 41039, 41351, 41539, 41879, 42083, 43451, 43499, 43787, 43951, 45179, 45863, 46663, 47111, 48247, 48563, 48679, 48883, 49207, 49367, 50147, 51431, 51563, 51683, 51899, 52103, 52783, 55667, 55711, 55787, 56383, 57787, 58031, 58099, 58231, 58567, 58979, 59023, 59407, 59611, 59743, 60923, 61091, 61751, 62071, 62327, 62743, 64451, 64499, 64567, 64951, 65003, 65183, 65899, 66403, 66587, 66883, 67003, 68147, 68219, 68279, 68903, 68947, 69899, 70019, 70183, 70619, 71879, 71987, 73331, 73547, 74831, 76231, 77491, 78347, 78571, 79279, 80779, 82139, 83207, 83243, 83663, 83843, 84319, 84391, 85411, 85427, 85451, 86539, 86627, 87511, 87587, 88223, 88427, 88811, 89599, 90439, 93739, 93923, 94463, 94547, 95471, 95723, 96731, 96763, 97607, 99083
類数 7 (86)
$\operatorname{Cl}(K) \cong C_7$ (86)
1087, 2251, 2467, 4139, 5039, 5743, 5827, 6151, 7919, 8311, 9539, 11447, 13003, 14827, 14947, 16127, 16339, 18587, 19087, 24023, 24359, 26251, 27211, 28151, 28751, 29851, 31391, 31771, 33179, 34883, 36263, 36551, 36779, 38711, 40507, 41851, 43319, 43991, 46559, 47819, 48539, 49711, 49939, 50923, 52879, 54371, 55163, 55903, 55927, 57271, 58271, 58631, 59539, 59747, 60259, 61631, 61723, 61979, 62423, 64891, 65563, 65927, 66239, 68539, 68899, 69491, 70639, 75767, 76387, 77263, 78691, 78787, 79259, 79379, 79943, 80279, 80803, 86287, 87407, 88339, 88651, 88867, 94327, 96211, 97127, 98887
類数 9 (54)
$\operatorname{Cl}(K) \cong C_3 \times C_3$ (3)
43063, 53507, 62687
$\operatorname{Cl}(K) \cong C_9$ (51)
3719, 5623, 11027, 12323, 13691, 14159, 21023, 23003, 24251, 25463, 29243, 29683, 29927, 30367, 35059, 35759, 36739, 38371, 40343, 40531, 40819, 42071, 42787, 43403, 43591, 44983, 45191, 45427, 47087, 47287, 47563, 47963, 53939, 58147, 58603, 60623, 62191, 65287, 67427, 71711, 72139, 76283, 77543, 77611, 81703, 82787, 87643, 89491, 95027, 95231, 98143
類数 11 (29)
$\operatorname{Cl}(K) \cong C_{11}$ (29)
8059, 10607, 12391, 14503, 19319, 23819, 25919, 41759, 41843, 42727, 43867, 44519, 49727, 52627, 53407, 57727, 59699, 64007, 65419, 71059, 72551, 74731, 78467, 84263, 84871, 95219, 95483, 97003, 98327
類数 13 (27)
$\operatorname{Cl}(K) \cong C_{13}$ (27)
4759, 8647, 11491, 12619, 19507, 22031, 32359, 40543, 40627, 42299, 44563, 55603, 56299, 58391, 58403, 59399, 66047, 70223, 73751, 78259, 83639, 85847, 89003, 91463, 98299, 99439, 99719
類数 15 (26)
$\operatorname{Cl}(K) \cong C_{15}$ (26)
9871, 15131, 16987, 17159, 21191, 21611, 22003, 23539, 28559, 35023, 40087, 42023, 42703, 47711, 50627, 51307, 51419, 54287, 61007, 62003, 64231, 67079, 68791, 75431, 84523, 93503
類数 17 (15)
$\operatorname{Cl}(K) \cong C_{17}$ (15)
14639, 18223, 32299, 36559, 40939, 56171, 57119, 59051, 60103, 62539, 74531, 81551, 82759, 85627, 89051
類数 19 (10)
$\operatorname{Cl}(K) \cong C_{19}$ (10)
33487, 37507, 57731, 66851, 71287, 83911, 90599, 91811, 93967, 98387
類数 21 (9)
$\operatorname{Cl}(K) \cong C_{21}$ (9)
13687, 28411, 36479, 52747, 60727, 65027, 84979, 89399, 93283
類数 23 (3)
$\operatorname{Cl}(K) \cong C_{23}$ (3)
65707, 66343, 74527
類数 25 (6)
$\operatorname{Cl}(K) \cong C_{25}$ (6)
24859, 46471, 68819, 69403, 88211, 97967
類数 27 (4)
$\operatorname{Cl}(K) \cong C_{27}$ (4)
30859, 31327, 56167, 90271
類数 29 (3)
$\operatorname{Cl}(K) \cong C_{29}$ (3)
49531, 61339, 99223
類数 33 (3)
$\operatorname{Cl}(K) \cong C_{33}$ (3)
55339, 81223, 95479
類数 37 (2)
$\operatorname{Cl}(K) \cong C_{37}$ (2)
53359, 77839
類数 39 (1)
$\operatorname{Cl}(K) \cong C_{39}$ (1)
81559
類数 45 (1)
$\operatorname{Cl}(K) \cong C_{45}$ (1)
91807
Magmaのコード
d:=5; // d として平方因子を持たない整数を取る
K := QuadraticField(d); // 二次体 K=Q(\sqrt{d}) を定義
Cl_K := ClassGroup(K); // K のイデアル類群 Cl(K) を計算
Cl_K; // イデアル類群 Cl(K) の出力
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