| 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 |
| 21.3.429...912.1 |
$x^{21} - x^{20} + 9 x^{19} - 7 x^{18} + 35 x^{17} - 27 x^{16} + 83 x^{15} - 61 x^{14} + 123 x^{13} - 111 x^{12} + 163 x^{11} - 225 x^{10} + 213 x^{9} - 253 x^{8} + 177 x^{7} - 167 x^{6} + 192 x^{5} - 180 x^{4} + 136 x^{3} - 76 x^{2} - 12 x + 4$ |
$21$ |
[3,9] |
$-\,2^{38}\cdot 3^{18}\cdot 7^{9}$ |
$3$ |
$20.6950697419$ |
$30.46083139496543$ |
|
|
|
$\PGL(2,7)$ (as 21T20) |
trivial |
$2$ |
$11$ |
$1790438.21987$ |
| 21.3.714...224.1 |
$x^{21} - 3 x^{20} - 4 x^{19} + 16 x^{18} - 9 x^{17} - 33 x^{16} + 52 x^{15} + 12 x^{14} - 136 x^{13} + 96 x^{12} + 180 x^{11} - 260 x^{10} - 144 x^{9} + 312 x^{8} + 44 x^{7} - 220 x^{6} + 15 x^{5} + 59 x^{4} - 16 x^{3} + 36 x^{2} - 31 x + 1$ |
$21$ |
[3,9] |
$-\,2^{64}\cdot 3^{18}$ |
$2$ |
$21.2024390858$ |
$31.63696468726565$ |
|
|
|
$\PGL(2,7)$ (as 21T20) |
trivial |
$2$ |
$11$ |
$3668230.07566$ |
| 21.5.279...288.1 |
$x^{21} - 2 x^{19} - 7 x^{18} - 5 x^{17} + 11 x^{16} + 17 x^{15} + 34 x^{14} - 32 x^{13} - 19 x^{12} - 120 x^{11} - 65 x^{10} - 276 x^{9} + 43 x^{8} + 539 x^{7} + 983 x^{6} + 606 x^{5} + 280 x^{4} + 121 x^{3} + 40 x^{2} - 13 x - 5$ |
$21$ |
[5,8] |
$2^{27}\cdot 181^{9}$ |
$2$ |
$22.6263811101$ |
$38.05259518088089$ |
|
|
|
$\PGL(2,7)$ (as 21T20) |
trivial |
$2$ |
$12$ |
$2296254.50823$ |
| 21.5.124...789.1 |
$x^{21} - 7 x^{20} + 22 x^{19} - 36 x^{18} + 16 x^{17} + 90 x^{16} - 359 x^{15} + 819 x^{14} - 1304 x^{13} + 1479 x^{12} - 987 x^{11} - 415 x^{10} + 2526 x^{9} - 4472 x^{8} + 5201 x^{7} - 4365 x^{6} + 2658 x^{5} - 1122 x^{4} + 291 x^{3} - 23 x^{2} - 4 x - 1$ |
$21$ |
[5,8] |
$1709^{9}$ |
$1$ |
$24.2918664051$ |
$41.340053217188775$ |
|
|
? |
$\PGL(2,7)$ (as 21T20) |
trivial |
$2$ |
$12$ |
$5054661.69478$ |
| 21.21.191...989.1 |
$x^{21} - 7 x^{20} - 74 x^{19} + 589 x^{18} + 2119 x^{17} - 20809 x^{16} - 26782 x^{15} + 399755 x^{14} + 51461 x^{13} - 4495452 x^{12} + 2641366 x^{11} + 29342562 x^{10} - 32803562 x^{9} - 100138483 x^{8} + 164165998 x^{7} + 116217006 x^{6} - 324673546 x^{5} + 109229003 x^{4} + 66372478 x^{3} - 20912877 x^{2} - 4643865 x + 27082$ |
$21$ |
[21,0] |
$107509^{9}$ |
$1$ |
$143.328810046$ |
$327.88565079917726$ |
|
|
? |
$\PGL(2,7)$ (as 21T20) |
$[2]$ |
$2$ |
$20$ |
$14849095216300000$ |
| 21.21.487...000.1 |
$x^{21} - 3 x^{20} - 114 x^{19} + 378 x^{18} + 4720 x^{17} - 18268 x^{16} - 92372 x^{15} + 440572 x^{14} + 854237 x^{13} - 5819847 x^{12} - 2098218 x^{11} + 43117146 x^{10} - 24499166 x^{9} - 172066922 x^{8} + 212267312 x^{7} + 313874448 x^{6} - 627347440 x^{5} - 93152336 x^{4} + 688442080 x^{3} - 275865888 x^{2} - 111827424 x + 53833824$ |
$21$ |
[21,0] |
$2^{49}\cdot 3^{8}\cdot 5^{8}\cdot 491^{8}$ |
$4$ |
$149.834557751$ |
$577.323005240646$ |
|
|
|
$\PGL(2,7)$ (as 21T20) |
trivial |
$2$ |
$20$ |
$2430053091190000000$ |
