| 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 |
Max $p$ |
| 27.27.117...601.1 |
$x^{27} - 171 x^{25} + 12555 x^{23} - 523308 x^{21} + 13786542 x^{19} - 74455 x^{18} - 241555041 x^{17} + 5446818 x^{16} + 2879538948 x^{15} - 163610928 x^{14} - 23447436363 x^{13} + 2612184666 x^{12} + 128773066383 x^{11} - 23924044755 x^{10} - 461913052263 x^{9} + 126068653854 x^{8} + 1017972390123 x^{7} - 361140735501 x^{6} - 1229645558349 x^{5} + 487411172820 x^{4} + 648969296859 x^{3} - 218553084711 x^{2} - 81582725718 x - 342703439$ |
$27$ |
(27, 0) |
$3^{66}\cdot 7^{18}\cdot 13^{12}$ |
$3$ |
$167.79577761928851$ |
$296.70440583414893$ |
|
|
? |
$C_9.C_3^2$ (as 27T28) |
trivial |
trivial |
$2$ |
$26$ |
$2618517662837614000000$ |
$27$ |
$13$ |
| 27.27.481...441.1 |
$x^{27} - 297 x^{25} - 333 x^{24} + 36072 x^{23} + 61137 x^{22} - 2408229 x^{21} - 4602204 x^{20} + 100240344 x^{19} + 188516023 x^{18} - 2781715041 x^{17} - 4662330939 x^{16} + 53254728639 x^{15} + 72347832963 x^{14} - 710593217208 x^{13} - 702999577791 x^{12} + 6541339948164 x^{11} + 4112713341072 x^{10} - 40351928688652 x^{9} - 13288812632739 x^{8} + 159103682965773 x^{7} + 20338350996222 x^{6} - 374458043565018 x^{5} - 13448413508607 x^{4} + 467216524743780 x^{3} + 21089835153009 x^{2} - 231674121736407 x - 25404456460733$ |
$27$ |
(27, 0) |
$3^{66}\cdot 7^{12}\cdot 13^{18}$ |
$3$ |
$192.54153546000558$ |
$296.70440583414893$ |
|
|
? |
$C_9.C_3^2$ (as 27T28) |
trivial |
trivial |
$2$ |
$26$ |
$19270877866665787000000$ |
$27$ |
$13$ |
| 27.27.565...209.11 |
$x^{27} - 297 x^{25} - 180 x^{24} + 36558 x^{23} + 38358 x^{22} - 2480523 x^{21} - 3319650 x^{20} + 104012604 x^{19} + 155921099 x^{18} - 2858638149 x^{17} - 4454857782 x^{16} + 53128287072 x^{15} + 81825196545 x^{14} - 676590154704 x^{13} - 991971371292 x^{12} + 5896656684999 x^{11} + 8000961948648 x^{10} - 34641989935247 x^{9} - 42599267805771 x^{8} + 133002999498051 x^{7} + 145547223780240 x^{6} - 315694281594060 x^{5} - 300360575166453 x^{4} + 415908907817568 x^{3} + 330937587298926 x^{2} - 230570178327909 x - 142504021189223$ |
$27$ |
(27, 0) |
$3^{66}\cdot 7^{18}\cdot 13^{18}$ |
$3$ |
$296.70440583414893$ |
$296.70440583414893$ |
|
|
? |
$C_9.C_3^2$ (as 27T28) |
$[3]$ |
$[3]$ |
$2$ |
$26$ |
$4049713353774727600000000$ |
$27$ |
$13$ |
| 27.27.565...209.12 |
$x^{27} - 297 x^{25} - 378 x^{24} + 37503 x^{23} + 90513 x^{22} - 2599194 x^{21} - 9080613 x^{20} + 106484463 x^{19} + 498317356 x^{18} - 2547457695 x^{17} - 16330694508 x^{16} + 30270621888 x^{15} + 326402938872 x^{14} + 8772267627 x^{13} - 3873478754556 x^{12} - 5058449596215 x^{11} + 24758238717396 x^{10} + 62122762900206 x^{9} - 59030518273902 x^{8} - 318361330780236 x^{7} - 129194717510394 x^{6} + 641062639490112 x^{5} + 753991064402733 x^{4} - 185376688343217 x^{3} - 600476957390412 x^{2} - 134154527840451 x + 72770998884921$ |
$27$ |
(27, 0) |
$3^{66}\cdot 7^{18}\cdot 13^{18}$ |
$3$ |
$296.70440583414893$ |
$296.70440583414893$ |
|
|
? |
$C_9.C_3^2$ (as 27T28) |
$[3]$ |
$[3]$ |
$2$ |
$26$ |
$8819548645435751000000000$ |
$27$ |
$13$ |
