| 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 |
| 18.2.393...424.1 |
$x^{18} - 3 x^{12} + 327 x^{6} - 1$ |
$18$ |
[2,8] |
$2^{18}\cdot 3^{36}$ |
$2$ |
$18.0$ |
$21.475876974362272$ |
|
|
? |
$C_3:S_4$ (as 18T37) |
trivial |
$2$ |
$9$ |
$39507.1018471$ |
| 18.2.200...000.1 |
$x^{18} - 8 x^{15} + 2 x^{12} + 76 x^{9} - 4 x^{6} - 32 x^{3} - 8$ |
$18$ |
[2,8] |
$2^{18}\cdot 3^{22}\cdot 5^{12}$ |
$3$ |
$22.3953677739$ |
$25.916910735084592$ |
|
|
? |
$C_3:S_4$ (as 18T37) |
$[3]$ |
$2$ |
$9$ |
$139689.480182$ |
| 18.6.481...096.1 |
$x^{18} - 9 x^{17} + 72 x^{16} - 372 x^{15} + 1584 x^{14} - 5292 x^{13} + 14172 x^{12} - 30510 x^{11} + 49905 x^{10} - 58939 x^{9} + 38484 x^{8} + 13770 x^{7} - 72978 x^{6} + 102312 x^{5} - 87048 x^{4} + 48672 x^{3} - 17280 x^{2} + 3456 x - 288$ |
$18$ |
[6,6] |
$2^{12}\cdot 3^{36}\cdot 23^{8}$ |
$3$ |
$57.5627514653$ |
$72.82824215856341$ |
|
|
|
$C_3:S_4$ (as 18T37) |
trivial |
$2$ |
$11$ |
$11618980288.3$ |
| 18.6.128...856.1 |
$x^{18} - 3 x^{17} - 42 x^{15} - 110 x^{14} + 332 x^{13} + 219 x^{12} + 1276 x^{11} - 3494 x^{10} - 12539 x^{9} + 24077 x^{8} + 22738 x^{7} + 12556 x^{6} - 9361 x^{5} - 277423 x^{4} + 8295 x^{3} + 548514 x^{2} + 236088 x - 60831$ |
$18$ |
[6,6] |
$2^{12}\cdot 7^{12}\cdot 197^{8}$ |
$3$ |
$60.7922369564$ |
$81.53019296306911$ |
|
|
|
$C_3:S_4$ (as 18T37) |
$[12]$ |
$2$ |
$11$ |
$1033559423.06$ |
| 18.18.250...656.1 |
$x^{18} - 3 x^{17} - 73 x^{16} + 214 x^{15} + 1688 x^{14} - 5008 x^{13} - 15868 x^{12} + 48500 x^{11} + 61275 x^{10} - 198755 x^{9} - 97997 x^{8} + 359116 x^{7} + 52726 x^{6} - 291004 x^{5} + 13708 x^{4} + 92466 x^{3} - 15181 x^{2} - 5415 x + 339$ |
$18$ |
[18,0] |
$2^{12}\cdot 3^{8}\cdot 2351^{8}$ |
$3$ |
$81.4845096803$ |
$133.31333890431625$ |
|
|
|
$C_3:S_4$ (as 18T37) |
$[3]$ |
$2$ |
$17$ |
$675507999150$ |
| 18.18.462...576.1 |
$x^{18} - 57 x^{16} + 1236 x^{14} - 13444 x^{12} + 82062 x^{10} - 297342 x^{8} + 651252 x^{6} - 843588 x^{4} + 593001 x^{2} - 173889$ |
$18$ |
[18,0] |
$2^{20}\cdot 3^{32}\cdot 47^{8}$ |
$3$ |
$84.3039532574$ |
$129.4626691961398$ |
|
|
|
$C_3:S_4$ (as 18T37) |
$[3]$ |
$2$ |
$17$ |
$851357937003$ |
