| Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Unit group torsion |
Unit group rank |
Regulator |
| 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 |
$2$ |
$17$ |
$13605448.556703938$ |
| 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 |
$2$ |
$17$ |
$932907706.847076$ |
| 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 |
$2$ |
$17$ |
$1816757010.3045344$ |
| 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 |
$2$ |
$17$ |
$2153441387.4033856$ |
| 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 |
$2$ |
$17$ |
$6360080771.530314$ |
| 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 |
$2$ |
$17$ |
$8061331492.668233$ |
