| Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Narrow class group |
Unit group torsion |
Unit group rank |
Regulator |
Unit signature rank |
| 27.3.244...329.1 |
$x^{27} - 6 x^{26} + 15 x^{25} - 18 x^{24} + 3 x^{23} + 18 x^{22} + 9 x^{21} - 129 x^{20} + 258 x^{19} - 249 x^{18} + 183 x^{17} - 375 x^{16} + 990 x^{15} - 1728 x^{14} + 2118 x^{13} - 2142 x^{12} + 2154 x^{11} - 2445 x^{10} + 3074 x^{9} - 3639 x^{8} + 3477 x^{7} - 2520 x^{6} + 1398 x^{5} - 624 x^{4} + 231 x^{3} - 66 x^{2} + 12 x - 1$ |
$27$ |
(3, 12) |
$3^{36}\cdot 7^{18}$ |
$2$ |
$15.83289626371223$ |
$16.82934070487988$ |
|
|
? |
$C_3^2:C_6$ (as 27T11) |
trivial |
trivial |
$2$ |
$14$ |
$181687.48822693975$ |
$3$ |
| 27.3.328...841.1 |
$x^{27} - 3 x^{26} - 3 x^{25} + 12 x^{24} - 24 x^{22} + 33 x^{21} + 36 x^{20} - 96 x^{19} - 55 x^{18} + 156 x^{17} + 54 x^{16} - 213 x^{15} - 99 x^{14} + 231 x^{13} + 189 x^{12} - 195 x^{11} - 219 x^{10} + 346 x^{9} + 447 x^{8} - 234 x^{7} - 549 x^{6} - 270 x^{5} - 27 x^{4} + 63 x^{3} + 69 x^{2} + 36 x + 1$ |
$27$ |
(3, 12) |
$3^{36}\cdot 23^{12}$ |
$2$ |
$17.433076844609083$ |
$22.05628309434856$ |
|
|
? |
$C_3^2:D_6$ (as 27T29) |
trivial |
trivial |
$2$ |
$14$ |
$720899.3580136786$ |
$3$ |
| 27.9.675...923.1 |
$x^{27} - 18 x^{23} - 18 x^{22} - 18 x^{21} + 18 x^{20} + 135 x^{19} + 219 x^{18} - 162 x^{17} + 324 x^{16} - 153 x^{15} - 81 x^{14} - 675 x^{13} - 630 x^{12} - 540 x^{11} - 513 x^{10} - 405 x^{9} + 27 x^{8} + 243 x^{7} + 387 x^{6} + 162 x^{5} - 81 x^{4} - 135 x^{3} + 27 x - 3$ |
$27$ |
(9, 9) |
$-\,3^{73}$ |
$1$ |
$19.498199223768765$ |
$19.898946404249138$ |
|
|
? |
$S_3\times C_9$ (as 27T12) |
trivial |
trivial |
$2$ |
$17$ |
$13605448.556703938$ |
$9$ |
| 27.3.120...336.1 |
$x^{27} - 3 x^{26} + 10 x^{25} - 13 x^{24} + 5 x^{23} + 15 x^{22} - 58 x^{21} - x^{20} - 22 x^{19} + 72 x^{18} - 240 x^{17} + 22 x^{16} + 671 x^{15} - 133 x^{14} - 326 x^{13} + 841 x^{12} + 1452 x^{11} + 1070 x^{10} + 1310 x^{9} + 1574 x^{8} + 912 x^{7} - 2 x^{6} - 292 x^{5} - 144 x^{4} - 11 x^{3} + 13 x^{2} + 4 x + 1$ |
$27$ |
(3, 12) |
$2^{66}\cdot 7^{18}$ |
$2$ |
$19.918186471121032$ |
$24.616776431006123$ |
|
|
|
$C_3^2:C_{12}$ (as 27T33) |
trivial |
trivial |
$2$ |
$14$ |
$6652859.434974616$ |
$3$ |
| 27.9.481...707.1 |
$x^{27} - 9 x^{25} + 36 x^{23} - 9 x^{22} - 96 x^{21} + 54 x^{20} - 33 x^{19} + 237 x^{18} + 162 x^{17} - 720 x^{16} + 459 x^{15} + 126 x^{14} + 723 x^{13} + 1167 x^{12} - 2412 x^{11} - 732 x^{10} - 124 x^{9} + 756 x^{8} + 2892 x^{7} - 2061 x^{6} - 1476 x^{5} + 1497 x^{4} - 81 x^{3} - 252 x^{2} + 84 x - 8$ |
$27$ |
(9, 9) |
$-\,3^{45}\cdot 7^{18}$ |
$2$ |
$22.834987833060737$ |
$24.27210940016998$ |
|
|
|
$S_3\times C_3^2$ (as 27T15) |
trivial |
trivial |
$2$ |
$17$ |
$120755171.39439106$ |
$9$ |
| 27.3.205...000.1 |
$x^{27} - 9 x^{26} + 39 x^{25} - 105 x^{24} + 180 x^{23} - 180 x^{22} + 60 x^{21} + 120 x^{20} - 135 x^{19} - 105 x^{18} - 585 x^{17} + 3075 x^{16} - 4200 x^{15} + 480 x^{14} + 5160 x^{13} - 5400 x^{12} + 1755 x^{11} + 1125 x^{10} - 3715 x^{9} + 2685 x^{8} - 300 x^{7} + 180 x^{6} + 540 x^{5} - 480 x^{4} + 75 x^{3} - 147 x^{2} - 27 x - 23$ |
$27$ |
(3, 12) |
$2^{24}\cdot 3^{30}\cdot 5^{24}$ |
$3$ |
$26.24361248771799$ |
$27.895256087472944$ |
|
|
? |
$C_3^3:S_3$ (as 27T51) |
trivial |
trivial |
$2$ |
$14$ |
$230439151.17926303$ |
$3$ |
| 27.9.218...752.1 |
$x^{27} - 18 x^{25} - 9 x^{24} + 135 x^{23} + 63 x^{22} - 642 x^{21} - 135 x^{20} + 2133 x^{19} + 336 x^{18} - 6156 x^{17} - 270 x^{16} + 15525 x^{15} - 4347 x^{14} - 26667 x^{13} + 16194 x^{12} + 26730 x^{11} - 24912 x^{10} - 13179 x^{9} + 19197 x^{8} + 693 x^{7} - 6957 x^{6} + 1917 x^{5} + 648 x^{4} - 507 x^{3} + 135 x^{2} - 18 x + 1$ |
$27$ |
(9, 9) |
$-\,2^{18}\cdot 3^{69}$ |
$2$ |
$26.302470182986912$ |
$26.84306578592094$ |
|
|
? |
$S_3\times C_9$ (as 27T12) |
trivial |
trivial |
$2$ |
$17$ |
$932907706.847076$ |
$9$ |
| 27.1.437...027.1 |
$x^{27} - x - 1$ |
$27$ |
(1, 13) |
$-\,23\cdot 3539\cdot 535391\cdot 10033918834509020645502251401$ |
$4$ |
$26.9860232809$ |
$2.091101070400067e+19$ |
|
|
✓ |
$S_{27}$ (as 27T2392) |
trivial |
trivial |
$2$ |
$13$ |
$289963252.83729804$ |
$1$ |
| 27.1.449...579.1 |
$x^{27} + x - 1$ |
$27$ |
(1, 13) |
$-\,151\cdot 144512650291622071\cdot 20602820982951053299$ |
$3$ |
$27.0137911008$ |
$2.120336312529795e+19$ |
|
|
✓ |
$S_{27}$ (as 27T2392) |
trivial |
trivial |
$2$ |
$13$ |
$314436524.8339556$ |
$1$ |
| 27.3.638...544.1 |
$x^{27} - 9 x^{25} + 36 x^{23} - 18 x^{22} - 144 x^{21} - 54 x^{20} + 423 x^{19} + 42 x^{18} - 405 x^{17} - 1062 x^{16} + 1188 x^{15} + 1872 x^{14} - 3258 x^{13} - 1980 x^{12} + 5724 x^{11} + 504 x^{10} - 9564 x^{9} + 7128 x^{8} + 3744 x^{7} - 8640 x^{6} + 4248 x^{5} + 1008 x^{4} - 1440 x^{3} + 288 x - 64$ |
$27$ |
(3, 12) |
$2^{56}\cdot 3^{46}$ |
$2$ |
$27.36728773462367$ |
$47.27452225037032$ |
|
|
|
$S_3\wr S_3$ (as 27T298) |
trivial |
trivial |
$2$ |
$14$ |
$19428990967.59157$ |
$3$ |
| 27.9.656...256.1 |
$x^{27} - 9 x^{26} + 27 x^{25} - 24 x^{24} - 9 x^{23} - 63 x^{22} + 273 x^{21} + 144 x^{20} - 2376 x^{19} + 4050 x^{18} - 396 x^{17} - 3465 x^{16} - 141 x^{15} + 441 x^{14} + 17019 x^{13} - 33225 x^{12} + 12852 x^{11} + 17379 x^{10} - 14277 x^{9} + 1188 x^{8} + 1143 x^{7} - 501 x^{6} + 891 x^{5} + 216 x^{4} + 129 x^{3} + 18 x^{2} + 9 x - 1$ |
$27$ |
(9, 9) |
$-\,2^{18}\cdot 3^{70}$ |
$2$ |
$27.394772269463456$ |
$34.51525023937336$ |
|
|
? |
$S_3\times C_9$ (as 27T12) |
trivial |
trivial |
$2$ |
$17$ |
$1816757010.3045344$ |
$9$ |
| 27.1.800...063.1 |
$x^{27} - 9 x^{26} + 41 x^{25} - 116 x^{24} + 234 x^{23} - 347 x^{22} + 380 x^{21} - 274 x^{20} + 118 x^{19} - 121 x^{18} + 474 x^{17} - 1071 x^{16} + 1556 x^{15} - 1531 x^{14} + 964 x^{13} - 256 x^{12} + 122 x^{11} - 700 x^{10} + 1761 x^{9} - 2620 x^{8} + 2840 x^{7} - 2483 x^{6} + 1678 x^{5} - 860 x^{4} + 463 x^{3} - 60 x^{2} + 44 x - 1$ |
$27$ |
(1, 13) |
$-\,983^{13}$ |
$1$ |
$27.5968238300111$ |
$31.352830813181765$ |
|
|
|
$D_{27}$ (as 27T8) |
trivial |
trivial |
$2$ |
$13$ |
$460164457.10213304$ |
$1$ |
| 27.3.995...896.1 |
$x^{27} - 12 x^{26} + 63 x^{25} - 192 x^{24} + 378 x^{23} - 480 x^{22} + 246 x^{21} + 624 x^{20} - 2346 x^{19} + 4608 x^{18} - 5598 x^{17} + 2304 x^{16} + 6564 x^{15} - 19824 x^{14} + 34248 x^{13} - 40080 x^{12} + 27606 x^{11} - 2784 x^{10} - 23718 x^{9} + 45552 x^{8} - 47970 x^{7} + 29328 x^{6} - 11814 x^{5} + 4704 x^{4} - 1779 x^{3} + 468 x^{2} - 105 x + 16$ |
$27$ |
(3, 12) |
$2^{82}\cdot 3^{30}$ |
$2$ |
$27.821079262374788$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$4994589101.470175$ |
$3$ |
| 27.3.133...409.1 |
$x^{27} - 9 x^{24} + 9 x^{21} + 6 x^{18} - 36 x^{15} - 315 x^{12} - 321 x^{9} - 117 x^{6} - 18 x^{3} - 1$ |
$27$ |
(3, 12) |
$3^{82}$ |
$1$ |
$28.121269452248747$ |
$28.699246901098466$ |
|
|
? |
$C_9:C_6$ (as 27T14) |
trivial |
trivial |
$2$ |
$14$ |
$777934687.5392165$ |
$3$ |
| 27.3.133...409.2 |
$x^{27} - 9 x^{24} + 9 x^{21} + 78 x^{18} - 252 x^{15} + 333 x^{12} - 249 x^{9} + 99 x^{6} - 18 x^{3} - 1$ |
$27$ |
(3, 12) |
$3^{82}$ |
$1$ |
$28.121269452248747$ |
$28.699246901098466$ |
|
|
? |
$C_3\times D_9$ (as 27T9) |
trivial |
trivial |
$2$ |
$14$ |
$669985731.8171878$ |
$3$ |
| 27.9.162...375.1 |
$x^{27} - 9 x^{25} + 54 x^{23} - 63 x^{22} - 267 x^{21} + 513 x^{20} + 468 x^{19} - 981 x^{18} - 972 x^{17} - 1107 x^{16} + 9846 x^{15} - 4860 x^{14} - 24624 x^{13} + 41403 x^{12} - 9909 x^{11} - 43317 x^{10} + 67377 x^{9} - 45441 x^{8} - 4905 x^{7} + 46368 x^{6} - 47520 x^{5} + 19584 x^{4} + 1146 x^{3} - 3483 x^{2} + 594 x + 107$ |
$27$ |
(9, 9) |
$-\,3^{69}\cdot 5^{9}$ |
$2$ |
$28.333477097878134$ |
$37.812070079830384$ |
|
|
? |
$S_3\times C_9$ (as 27T12) |
trivial |
trivial |
$2$ |
$17$ |
$2153441387.4033856$ |
$9$ |
| 27.3.197...000.1 |
$x^{27} + 18 x^{25} + 135 x^{23} + 549 x^{21} + 1323 x^{19} - 252 x^{18} + 1944 x^{17} - 1566 x^{16} + 2133 x^{15} - 3726 x^{14} + 2430 x^{13} - 4509 x^{12} + 2430 x^{11} - 2754 x^{10} - 108 x^{9} - 729 x^{8} - 1782 x^{7} - 567 x^{6} - 972 x^{5} - 243 x^{4} - 162 x^{3} - 27$ |
$27$ |
(3, 12) |
$2^{12}\cdot 3^{66}\cdot 5^{6}$ |
$3$ |
$28.537385187853925$ |
$66.93403046590726$ |
|
|
|
$S_3\wr S_3$ (as 27T298) |
trivial |
trivial |
$2$ |
$14$ |
$3419872326.2495065$ |
$3$ |
| 27.3.291...936.1 |
$x^{27} - 9 x^{25} - 9 x^{24} + 36 x^{23} + 45 x^{22} - 39 x^{21} - 180 x^{20} + 18 x^{19} + 471 x^{18} + 783 x^{17} - 702 x^{16} - 1080 x^{15} - 459 x^{14} + 2907 x^{13} + 3024 x^{12} - 459 x^{11} - 3411 x^{10} - 906 x^{9} + 4455 x^{8} + 3852 x^{7} - 486 x^{6} - 5616 x^{5} - 1404 x^{4} + 2304 x^{3} + 1728 x^{2} - 768$ |
$27$ |
(3, 12) |
$2^{36}\cdot 3^{60}$ |
$2$ |
$28.949614336814307$ |
$33.16271107577659$ |
|
|
|
$C_3:S_3^2$ (as 27T35) |
$[3]$ |
$[3]$ |
$2$ |
$14$ |
$10259483992.714678$ |
$3$ |
| 27.1.321...296.1 |
$x^{27} - 10 x^{26} + 38 x^{25} - 52 x^{24} - 84 x^{23} + 458 x^{22} - 715 x^{21} + 40 x^{20} + 1948 x^{19} - 3756 x^{18} + 1448 x^{17} + 4952 x^{16} - 7109 x^{15} + 94 x^{14} + 4978 x^{13} - 2352 x^{12} - 1300 x^{11} + 2422 x^{10} + 639 x^{9} - 3848 x^{8} - 28 x^{7} + 1728 x^{6} + 572 x^{5} - 368 x^{4} - 244 x^{3} + 48 x^{2} + 32 x - 16$ |
$27$ |
(1, 13) |
$-\,2^{18}\cdot 419^{13}$ |
$2$ |
$29.055888235511343$ |
$32.49328915040618$ |
|
|
|
$D_{27}$ (as 27T8) |
trivial |
trivial |
$2$ |
$13$ |
$2806458409.197778$ |
$1$ |
| 27.3.485...000.1 |
$x^{27} - 3 x^{26} - 9 x^{25} + 17 x^{24} - 6 x^{23} - 86 x^{22} + 42 x^{21} - 26 x^{20} + 2 x^{19} + 222 x^{18} - 620 x^{17} - 136 x^{16} + 1722 x^{15} - 1022 x^{14} - 1740 x^{13} + 3140 x^{12} + 278 x^{11} - 3510 x^{10} + 1746 x^{9} + 1702 x^{8} - 1886 x^{7} - 6 x^{6} + 1068 x^{5} - 272 x^{4} - 230 x^{3} + 166 x^{2} + 10 x - 58$ |
$27$ |
(3, 12) |
$2^{52}\cdot 3^{24}\cdot 5^{18}$ |
$3$ |
$29.501850506318394$ |
$51.05223872742641$ |
|
|
? |
$\He_3:\GL(2,3)$ (as 27T294) |
trivial |
trivial |
$2$ |
$14$ |
$5251358132.129515$ |
$3$ |
| 27.3.708...816.1 |
$x^{27} - 3 x^{26} + 3 x^{25} - 25 x^{24} + 60 x^{23} - 60 x^{22} + 152 x^{21} - 216 x^{20} + 114 x^{19} + 234 x^{18} - 78 x^{17} + 762 x^{16} - 1152 x^{15} + 816 x^{14} - 96 x^{13} + 352 x^{12} - 984 x^{11} + 1128 x^{10} - 872 x^{9} + 696 x^{8} - 864 x^{7} + 480 x^{6} - 192 x^{5} + 192 x^{4} - 336 x^{3} + 240 x^{2} - 144 x + 48$ |
$27$ |
(3, 12) |
$2^{88}\cdot 3^{28}$ |
$2$ |
$29.91763180173999$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$26800740160.001953$ |
$3$ |
| 27.7.769...000.1 |
$x^{27} - 3 x^{26} - 12 x^{25} + 54 x^{24} + 42 x^{23} - 588 x^{22} + 1104 x^{21} + 576 x^{20} - 5730 x^{19} + 9174 x^{18} - 222 x^{17} - 23034 x^{16} + 40404 x^{15} - 22632 x^{14} - 33780 x^{13} + 92028 x^{12} - 106221 x^{11} + 66099 x^{10} - 4728 x^{9} - 36054 x^{8} + 42234 x^{7} - 28380 x^{6} + 13176 x^{5} - 4332 x^{4} + 936 x^{3} - 96 x^{2} - 6 x + 2$ |
$27$ |
(7, 10) |
$2^{52}\cdot 3^{42}\cdot 5^{6}$ |
$3$ |
$30.010743793252555$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$16$ |
$32838878089.412144$ |
$7$ |
| 27.3.886...000.1 |
$x^{27} - 9 x^{26} + 36 x^{25} - 51 x^{24} - 114 x^{23} + 651 x^{22} - 1518 x^{21} + 2583 x^{20} - 3606 x^{19} + 3839 x^{18} - 2586 x^{17} + 459 x^{16} + 1188 x^{15} - 1815 x^{14} + 930 x^{13} + 1485 x^{12} - 4410 x^{11} + 7077 x^{10} - 8306 x^{9} + 7125 x^{8} - 4590 x^{7} + 1965 x^{6} - 162 x^{5} - 423 x^{4} + 387 x^{3} - 216 x^{2} + 54 x - 18$ |
$27$ |
(3, 12) |
$2^{18}\cdot 3^{46}\cdot 5^{18}$ |
$3$ |
$30.167541264200818$ |
$30.78757581020732$ |
|
|
|
$C_3^2:S_3$ (as 27T6) |
$[3]$ |
$[3]$ |
$2$ |
$14$ |
$14461249563.554726$ |
$3$ |
| 27.3.952...000.1 |
$x^{27} - 9 x^{25} - 9 x^{24} + 18 x^{23} + 72 x^{22} + 60 x^{21} - 126 x^{20} - 252 x^{19} - 162 x^{18} + 108 x^{17} + 126 x^{16} - 216 x^{15} + 450 x^{14} + 1692 x^{13} + 2670 x^{12} + 1899 x^{11} - 900 x^{10} - 1461 x^{9} - 3591 x^{8} - 6354 x^{7} - 990 x^{6} + 4392 x^{5} + 2448 x^{4} - 435 x^{3} - 720 x^{2} - 225 x - 25$ |
$27$ |
(3, 12) |
$2^{24}\cdot 3^{54}\cdot 5^{10}$ |
$3$ |
$30.24840694808679$ |
$39.61106018062899$ |
|
|
? |
$C_3^3:S_4$ (as 27T211) |
trivial |
trivial |
$2$ |
$14$ |
$7929164662.056825$ |
$3$ |
| 27.3.108...000.1 |
$x^{27} - 3 x^{26} + 6 x^{25} - 24 x^{24} + 48 x^{23} - 168 x^{22} + 432 x^{21} - 912 x^{20} + 1881 x^{19} - 3339 x^{18} + 5010 x^{17} - 6228 x^{16} + 4974 x^{15} + 678 x^{14} - 7944 x^{13} + 10788 x^{12} - 6252 x^{11} - 1380 x^{10} + 6072 x^{9} - 6312 x^{8} + 3396 x^{7} + 324 x^{6} - 2112 x^{5} + 1416 x^{4} - 156 x^{3} - 252 x^{2} + 120 x - 16$ |
$27$ |
(3, 12) |
$2^{62}\cdot 3^{36}\cdot 5^{6}$ |
$3$ |
$30.390819048885994$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$9454549156.39464$ |
$3$ |
| 27.3.128...000.1 |
$x^{27} - 5 x^{26} + 11 x^{25} - 27 x^{24} + 46 x^{23} - 74 x^{22} + 120 x^{21} - 136 x^{20} + 173 x^{19} - 421 x^{18} + 473 x^{17} - 137 x^{16} + 1820 x^{15} - 868 x^{14} + 664 x^{13} - 2904 x^{12} + 1215 x^{11} - 2067 x^{10} - 1247 x^{9} - 337 x^{8} + 3998 x^{7} + 2582 x^{6} - 2528 x^{5} - 2064 x^{4} + 1443 x^{3} + 1693 x^{2} - 909 x - 2243$ |
$27$ |
(3, 12) |
$2^{68}\cdot 3^{6}\cdot 5^{24}$ |
$3$ |
$30.583221823651705$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$66888533375.85289$ |
$3$ |
| 27.3.131...000.1 |
$x^{27} - 9 x^{26} + 33 x^{25} - 41 x^{24} - 111 x^{23} + 597 x^{22} - 1314 x^{21} + 1746 x^{20} - 1209 x^{19} - 463 x^{18} + 2265 x^{17} - 3507 x^{16} + 4726 x^{15} - 6558 x^{14} + 8271 x^{13} - 2855 x^{12} - 13755 x^{11} + 25959 x^{10} - 17496 x^{9} - 1668 x^{8} + 13713 x^{7} - 14007 x^{6} + 7287 x^{5} - 1335 x^{4} + 331 x^{3} - 255 x^{2} + 42 x - 4$ |
$27$ |
(3, 12) |
$2^{30}\cdot 3^{30}\cdot 5^{24}$ |
$3$ |
$30.613936070305446$ |
$37.9595855361366$ |
|
|
|
$C_3^3:D_6$ (as 27T128) |
trivial |
trivial |
$2$ |
$14$ |
$56929818889.756966$ |
$3$ |
| 27.1.149...191.1 |
$x^{27} - 4 x^{26} + 14 x^{25} - 10 x^{24} + 37 x^{22} - 3 x^{21} - x^{20} + 139 x^{19} - 26 x^{18} + 76 x^{17} + 144 x^{16} - 117 x^{15} - 101 x^{14} + 12 x^{13} - 340 x^{12} + x^{11} + 205 x^{10} + 307 x^{9} + 774 x^{8} + 909 x^{7} + 892 x^{6} + 864 x^{5} + 521 x^{4} + 327 x^{3} + 86 x^{2} + 71 x + 1$ |
$27$ |
(1, 13) |
$-\,1231^{13}$ |
$1$ |
$30.75402942005746$ |
$35.08560958569767$ |
|
|
|
$D_{27}$ (as 27T8) |
trivial |
trivial |
$2$ |
$13$ |
$5105816638.575763$ |
$1$ |
| 27.3.163...864.1 |
$x^{27} - 6 x^{24} - 51 x^{21} + 36 x^{20} - 36 x^{19} + 576 x^{18} - 72 x^{17} + 396 x^{16} - 2727 x^{15} - 3600 x^{14} - 7794 x^{13} - 6246 x^{12} - 5355 x^{11} + 2124 x^{10} + 9909 x^{9} + 14238 x^{8} + 14238 x^{7} + 8964 x^{6} + 657 x^{5} - 2016 x^{4} - 1188 x^{3} - 144 x^{2} + 180 x + 96$ |
$27$ |
(3, 12) |
$2^{48}\cdot 3^{54}$ |
$2$ |
$30.860783382710622$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$29005138815.170612$ |
$3$ |
| 27.9.177...912.1 |
$x^{27} - 18 x^{25} + 189 x^{23} - 63 x^{22} - 1062 x^{21} + 648 x^{20} + 4464 x^{19} - 2679 x^{18} - 14499 x^{17} + 7092 x^{16} + 36963 x^{15} - 21411 x^{14} - 74187 x^{13} + 24255 x^{12} + 61398 x^{11} - 36351 x^{10} - 17520 x^{9} + 66663 x^{8} + 42507 x^{7} - 11583 x^{6} - 18603 x^{5} - 10539 x^{4} - 6327 x^{3} - 2835 x^{2} - 639 x - 53$ |
$27$ |
(9, 9) |
$-\,2^{18}\cdot 3^{73}$ |
$2$ |
$30.95146195929607$ |
$31.5876084551639$ |
|
|
? |
$S_3\times C_9$ (as 27T12) |
trivial |
trivial |
$2$ |
$17$ |
$6360080771.530314$ |
$9$ |
| 27.9.177...912.2 |
$x^{27} - 9 x^{26} + 36 x^{25} - 54 x^{24} - 180 x^{23} + 1296 x^{22} - 3699 x^{21} + 5184 x^{20} + 1737 x^{19} - 28278 x^{18} + 81630 x^{17} - 152073 x^{16} + 206037 x^{15} - 206640 x^{14} + 143748 x^{13} - 36954 x^{12} - 70929 x^{11} + 125325 x^{10} - 106329 x^{9} + 53154 x^{8} - 10548 x^{7} - 9594 x^{6} + 13185 x^{5} - 9180 x^{4} + 4068 x^{3} - 1053 x^{2} - 207 x + 163$ |
$27$ |
(9, 9) |
$-\,2^{18}\cdot 3^{73}$ |
$2$ |
$30.95146195929607$ |
$31.5876084551639$ |
|
|
? |
$S_3\times C_9$ (as 27T12) |
trivial |
trivial |
$2$ |
$17$ |
$8061331492.668233$ |
$9$ |
| 27.27.215...729.1 |
$x^{27} - 10 x^{26} + 10 x^{25} + 218 x^{24} - 783 x^{23} - 754 x^{22} + 7599 x^{21} - 5094 x^{20} - 31037 x^{19} + 43000 x^{18} + 68238 x^{17} - 135637 x^{16} - 91409 x^{15} + 242376 x^{14} + 86222 x^{13} - 272022 x^{12} - 72196 x^{11} + 195156 x^{10} + 58033 x^{9} - 84568 x^{8} - 34757 x^{7} + 17820 x^{6} + 11437 x^{5} - 146 x^{4} - 1325 x^{3} - 338 x^{2} - 32 x - 1$ |
$27$ |
(27, 0) |
$7^{24}\cdot 13^{18}$ |
$2$ |
$31.176485742782422$ |
$31.176485742782422$ |
|
✓ |
? |
$C_9:C_3$ (as 27T5) |
trivial |
trivial |
$2$ |
$26$ |
$236492765885.6633$ |
$27$ |
| 27.1.252...624.1 |
$x^{27} - 9 x^{26} + 45 x^{25} - 154 x^{24} + 444 x^{23} - 1140 x^{22} + 2734 x^{21} - 5824 x^{20} + 11115 x^{19} - 18567 x^{18} + 29531 x^{17} - 43306 x^{16} + 56712 x^{15} - 56676 x^{14} + 52570 x^{13} - 50872 x^{12} + 59685 x^{11} - 40721 x^{10} + 10893 x^{9} + 10490 x^{8} - 13378 x^{7} + 7126 x^{6} - 3604 x^{5} - 204 x^{4} + 1348 x^{3} - 376 x^{2} + 8 x + 8$ |
$27$ |
(1, 13) |
$-\,2^{18}\cdot 491^{13}$ |
$2$ |
$31.361192438338826$ |
$35.174457650357105$ |
|
|
|
$D_{27}$ (as 27T8) |
$[2]$ |
$[2]$ |
$2$ |
$13$ |
$4486112564.467699$ |
$1$ |
| 27.3.453...824.1 |
$x^{27} - 3 x^{26} + 6 x^{25} - 18 x^{24} + 72 x^{23} - 204 x^{22} + 438 x^{21} - 726 x^{20} + 729 x^{19} + 449 x^{18} - 3888 x^{17} + 8532 x^{16} - 8376 x^{15} - 8784 x^{14} + 55152 x^{13} - 132288 x^{12} + 221472 x^{11} - 287712 x^{10} + 297440 x^{9} - 244608 x^{8} + 159168 x^{7} - 84288 x^{6} + 41328 x^{5} - 22800 x^{4} + 13176 x^{3} - 6024 x^{2} + 1728 x - 224$ |
$27$ |
(3, 12) |
$2^{78}\cdot 3^{36}$ |
$2$ |
$32.04821774291616$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$112106209033.01419$ |
$3$ |
| 27.1.786...199.1 |
$x^{27} - x^{26} + 6 x^{25} - 31 x^{24} + 97 x^{23} - 269 x^{22} + 599 x^{21} - 994 x^{20} + 1307 x^{19} - 1298 x^{18} + 592 x^{17} + 817 x^{16} - 2461 x^{15} + 4001 x^{14} - 4903 x^{13} + 4315 x^{12} - 2279 x^{11} - 512 x^{10} + 3872 x^{9} - 6814 x^{8} + 7826 x^{7} - 7115 x^{6} + 5689 x^{5} - 3842 x^{4} + 1951 x^{3} - 627 x^{2} + 101 x - 1$ |
$27$ |
(1, 13) |
$-\,1399^{13}$ |
$1$ |
$32.707923524616376$ |
$37.403208418530085$ |
|
|
|
$D_{27}$ (as 27T8) |
$[2]$ |
$[2]$ |
$2$ |
$13$ |
$6401978481.787342$ |
$1$ |
| 27.3.828...624.1 |
$x^{27} + 9 x^{25} - 12 x^{24} + 18 x^{23} - 108 x^{22} - 78 x^{21} - 252 x^{20} - 549 x^{19} - 180 x^{18} - 2421 x^{17} - 1440 x^{16} - 4722 x^{15} - 4032 x^{14} + 4302 x^{13} + 672 x^{12} + 18828 x^{11} + 5832 x^{10} - 7284 x^{9} - 12528 x^{8} - 55944 x^{7} - 31872 x^{6} - 24768 x^{5} - 2016 x^{4} + 20640 x^{3} + 4608 x^{2} + 2880 x - 15488$ |
$27$ |
(3, 12) |
$2^{44}\cdot 3^{58}$ |
$2$ |
$32.77136568082241$ |
$42.36304129546679$ |
|
|
|
$C_3^3:S_4$ (as 27T211) |
$[3]$ |
$[3]$ |
$2$ |
$14$ |
$68475153223.48233$ |
$3$ |
| 27.3.919...336.1 |
$x^{27} - 9 x^{26} + 27 x^{25} - 19 x^{24} + 6 x^{23} - 378 x^{22} + 1324 x^{21} - 1272 x^{20} + 27 x^{19} - 5467 x^{18} + 21285 x^{17} - 25389 x^{16} + 8080 x^{15} - 28104 x^{14} + 120960 x^{13} - 179344 x^{12} + 107400 x^{11} - 8856 x^{10} + 10008 x^{9} - 60120 x^{8} + 70560 x^{7} - 49536 x^{6} + 29952 x^{5} - 17280 x^{4} + 9072 x^{3} - 3888 x^{2} + 1296 x - 144$ |
$27$ |
(3, 12) |
$2^{60}\cdot 3^{48}$ |
$2$ |
$32.89812223106083$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$76916706852.12234$ |
$3$ |
| 27.7.129...616.1 |
$x^{27} - 90 x^{19} - 336 x^{18} + 1080 x^{15} - 1215 x^{11} + 3312 x^{10} - 2544 x^{9} + 1944 x^{7} - 1728 x^{6} - 324 x^{3} - 144 x - 64$ |
$27$ |
(7, 10) |
$2^{70}\cdot 3^{42}$ |
$2$ |
$33.31476509412915$ |
|
|
|
|
$\PSp(4,3)$ (as 27T993) |
trivial |
trivial |
$2$ |
$16$ |
$322730154520.6651$ |
$7$ |
| 27.1.149...032.1 |
$x^{27} - 7 x^{26} + 21 x^{25} - 50 x^{24} + 110 x^{23} - 244 x^{22} + 734 x^{21} - 2140 x^{20} + 5743 x^{19} - 13591 x^{18} + 28791 x^{17} - 54584 x^{16} + 93410 x^{15} - 138060 x^{14} + 176010 x^{13} - 189870 x^{12} + 169389 x^{11} - 127135 x^{10} + 89765 x^{9} - 72912 x^{8} + 70628 x^{7} - 65648 x^{6} + 50256 x^{5} - 29876 x^{4} + 13340 x^{3} - 4224 x^{2} + 848 x - 80$ |
$27$ |
(1, 13) |
$-\,2^{18}\cdot 563^{13}$ |
$2$ |
$33.49696971150064$ |
$37.66525059231157$ |
|
|
|
$D_{27}$ (as 27T8) |
$[2]$ |
$[2]$ |
$2$ |
$13$ |
$14887194175.434155$ |
$1$ |
| 27.3.181...296.1 |
$x^{27} - 3 x^{26} - 15 x^{25} + 45 x^{24} + 78 x^{23} - 210 x^{22} - 198 x^{21} + 306 x^{20} + 231 x^{19} - 493 x^{18} + 1887 x^{17} + 3 x^{16} - 8556 x^{15} + 12804 x^{14} - 7536 x^{13} - 6672 x^{12} + 20184 x^{11} - 10560 x^{10} - 11760 x^{9} + 30192 x^{8} - 53700 x^{7} + 65796 x^{6} - 58956 x^{5} + 51156 x^{4} - 36306 x^{3} + 18390 x^{2} - 8994 x + 2870$ |
$27$ |
(3, 12) |
$2^{80}\cdot 3^{36}$ |
$2$ |
$33.736684588919076$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$226969346121.17557$ |
$3$ |
| 27.3.261...824.1 |
$x^{27} - 9 x^{26} + 36 x^{25} - 78 x^{24} + 90 x^{23} - 36 x^{22} - 72 x^{21} + 324 x^{20} - 648 x^{19} + 412 x^{18} + 198 x^{17} - 198 x^{16} + 708 x^{15} - 864 x^{14} - 792 x^{13} + 216 x^{12} + 1026 x^{11} - 3114 x^{10} + 6088 x^{9} - 2124 x^{8} - 1116 x^{7} + 3456 x^{6} - 8136 x^{5} + 4536 x^{4} + 1512 x^{3} - 2664 x^{2} + 1692 x - 636$ |
$27$ |
(3, 12) |
$2^{52}\cdot 3^{54}$ |
$2$ |
$34.198257855009565$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$163300072007.99246$ |
$3$ |
| 27.3.393...000.1 |
$x^{27} - 6 x^{25} - 22 x^{24} + 6 x^{23} + 156 x^{22} + 166 x^{21} - 540 x^{20} - 1284 x^{19} + 764 x^{18} + 5586 x^{17} + 4776 x^{16} - 8664 x^{15} - 21792 x^{14} - 10272 x^{13} + 20704 x^{12} + 28602 x^{11} - 7008 x^{10} - 46852 x^{9} - 36588 x^{8} + 37356 x^{7} + 139024 x^{6} + 207180 x^{5} + 197280 x^{4} + 124824 x^{3} + 50112 x^{2} + 11316 x + 1048$ |
$27$ |
(3, 12) |
$2^{52}\cdot 3^{28}\cdot 5^{18}$ |
$3$ |
$34.71633642477372$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$134255356693.02907$ |
$3$ |
| 27.1.476...207.1 |
$x^{27} - 3 x^{26} - 11 x^{25} + 17 x^{24} + 38 x^{23} + x^{22} + 157 x^{21} + 384 x^{20} + 124 x^{19} - 139 x^{18} + 318 x^{17} + 1284 x^{16} + 1757 x^{15} - 73 x^{14} - 3289 x^{13} - 3942 x^{12} - 2202 x^{11} - 2459 x^{10} - 3813 x^{9} - 1971 x^{8} + 2174 x^{7} + 7281 x^{6} + 9348 x^{5} + 4444 x^{4} - 209 x^{3} - 325 x^{2} + 132 x + 1$ |
$27$ |
(1, 13) |
$-\,1607^{13}$ |
$1$ |
$34.96530122267493$ |
$40.08740450565489$ |
|
|
|
$D_{27}$ (as 27T8) |
trivial |
trivial |
$2$ |
$13$ |
$16112744313.230444$ |
$1$ |
| 27.3.573...096.1 |
$x^{27} - 96 x^{21} - 180 x^{19} - 672 x^{18} - 1248 x^{15} + 4608 x^{13} - 5376 x^{12} - 4860 x^{11} + 13248 x^{10} - 15296 x^{9} + 21312 x^{7} - 10752 x^{6} - 10368 x^{5} + 4608 x^{4} + 1632 x^{3} - 1152 x - 512$ |
$27$ |
(3, 12) |
$2^{88}\cdot 3^{32}$ |
$2$ |
$35.20560754110227$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$367749295231.59753$ |
$3$ |
| 27.3.609...000.1 |
$x^{27} + 9 x^{25} - 27 x^{24} + 18 x^{23} + 18 x^{22} + 144 x^{21} - 144 x^{20} - 288 x^{19} + 120 x^{18} + 1944 x^{17} - 2304 x^{16} + 1944 x^{15} - 720 x^{14} - 288 x^{13} + 720 x^{12} + 8352 x^{11} - 7488 x^{10} + 8688 x^{9} - 1296 x^{8} - 5472 x^{7} + 6912 x^{6} - 5760 x^{5} + 1152 x^{4} + 576 x^{3} - 576 x^{2} + 576 x + 64$ |
$27$ |
(3, 12) |
$2^{30}\cdot 3^{54}\cdot 5^{10}$ |
$3$ |
$35.28564510585922$ |
|
|
|
? |
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$43139206225.47275$ |
$3$ |
| 27.3.653...056.1 |
$x^{27} - 3 x^{24} + 276 x^{21} - 3076 x^{18} + 10032 x^{15} - 12720 x^{12} + 9568 x^{9} + 792 x^{6} + 135 x^{3} + 27$ |
$27$ |
(3, 12) |
$2^{66}\cdot 3^{46}$ |
$2$ |
$35.37727270014978$ |
$64.35491690563994$ |
|
|
|
$\He_3:\GL(2,3)$ (as 27T294) |
trivial |
trivial |
$2$ |
$14$ |
$363833734516.5203$ |
$3$ |
| 27.3.970...752.1 |
$x^{27} - 15 x^{24} - 3 x^{21} + 526 x^{18} + 1350 x^{15} + 273 x^{12} - 551 x^{9} - 297 x^{6} - 42 x^{3} - 1$ |
$27$ |
(3, 12) |
$2^{18}\cdot 3^{57}\cdot 11^{9}$ |
$3$ |
$35.89827176374774$ |
$56.904383806758446$ |
|
|
? |
$C_3\times S_3^2$ (as 27T36) |
trivial |
trivial |
$2$ |
$14$ |
$15120935926.540888$ |
$3$ |
| 27.27.105...721.1 |
$x^{27} - 45 x^{25} - 18 x^{24} + 837 x^{23} + 684 x^{22} - 8262 x^{21} - 10287 x^{20} + 45702 x^{19} + 79026 x^{18} - 134379 x^{17} - 333945 x^{16} + 151686 x^{15} + 784917 x^{14} + 175518 x^{13} - 983601 x^{12} - 695061 x^{11} + 544437 x^{10} + 756045 x^{9} + 21222 x^{8} - 337599 x^{7} - 142434 x^{6} + 39528 x^{5} + 42858 x^{4} + 7524 x^{3} - 2268 x^{2} - 1008 x - 107$ |
$27$ |
(27, 0) |
$3^{72}\cdot 19^{6}$ |
$2$ |
$36.015613145770395$ |
$140.73019518097922$ |
|
|
? |
$C_3^3:C_9$ (as 27T98) |
trivial |
$[2, 2]$ |
$2$ |
$26$ |
$2652309295077.3276$ |
$25$ |
| 27.1.154...079.1 |
$x^{27} - 6 x^{26} + 31 x^{25} - 80 x^{24} + 117 x^{23} - 51 x^{22} - 243 x^{21} + 807 x^{20} - 1350 x^{19} + 966 x^{18} + 1935 x^{17} - 9345 x^{16} + 23103 x^{15} - 43548 x^{14} + 68751 x^{13} - 94575 x^{12} + 115269 x^{11} - 125268 x^{10} + 121785 x^{9} - 105567 x^{8} + 81027 x^{7} - 54357 x^{6} + 31122 x^{5} - 14664 x^{4} + 5393 x^{3} - 1350 x^{2} + 179 x - 1$ |
$27$ |
(1, 13) |
$-\,1759^{13}$ |
$1$ |
$36.52038978810907$ |
$41.94043395102154$ |
|
|
|
$D_{27}$ (as 27T8) |
trivial |
trivial |
$2$ |
$13$ |
$71844122831.24568$ |
$1$ |
| 27.3.197...000.1 |
$x^{27} - 72 x^{22} - 72 x^{21} - 450 x^{19} - 840 x^{18} + 324 x^{17} + 864 x^{16} - 5832 x^{15} - 3240 x^{14} + 11700 x^{13} - 2232 x^{12} - 37503 x^{11} + 19800 x^{10} + 46524 x^{9} - 71280 x^{8} - 18360 x^{7} + 103680 x^{6} - 83268 x^{5} - 2232 x^{4} + 46692 x^{3} - 33696 x^{2} + 10296 x - 1648$ |
$27$ |
(3, 12) |
$2^{60}\cdot 3^{42}\cdot 5^{6}$ |
$3$ |
$36.85282899997973$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$265591649463.0986$ |
$3$ |
