| 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 $p$ |
| 27.27.284...649.3 |
$x^{27} - 171 x^{25} - 114 x^{24} + 11970 x^{23} + 12483 x^{22} - 453663 x^{21} - 538821 x^{20} + 10425870 x^{19} + 11862935 x^{18} - 154489950 x^{17} - 141607665 x^{16} + 1529020554 x^{15} + 844655526 x^{14} - 10215622764 x^{13} - 918820449 x^{12} + 44892716355 x^{11} - 19060261749 x^{10} - 118314463817 x^{9} + 117959236929 x^{8} + 138373678629 x^{7} - 272660806212 x^{6} + 54829227276 x^{5} + 187694093196 x^{4} - 190450670424 x^{3} + 80727665823 x^{2} - 16167225438 x + 1212239083$ |
$27$ |
(27, 0) |
$3^{54}\cdot 19^{24}$ |
$2$ |
$123.28560676835092$ |
$145.0764791851203$ |
|
|
? |
$C_3.\He_3$ (as 27T20) |
$[3]$ |
$[3]$ |
$2$ |
$26$ |
$24521012470764184000$ |
$27$ |
$19$ |
| 27.27.251...409.3 |
$x^{27} - 333 x^{25} - 333 x^{24} + 35298 x^{23} + 53946 x^{22} - 1819068 x^{21} - 3588075 x^{20} + 52704576 x^{19} + 127734434 x^{18} - 907552539 x^{17} - 2673287370 x^{16} + 9278172984 x^{15} + 34236633762 x^{14} - 51653439189 x^{13} - 270237162441 x^{12} + 93428282529 x^{11} + 1280114382555 x^{10} + 527095739526 x^{9} - 3356233487919 x^{8} - 3330019648665 x^{7} + 3845500135071 x^{6} + 6646340649357 x^{5} + 111023372826 x^{4} - 4196847578151 x^{3} - 2127671869662 x^{2} - 113904515466 x + 74779074553$ |
$27$ |
(27, 0) |
$3^{54}\cdot 37^{24}$ |
$2$ |
$222.9459551667726$ |
$262.35190848297987$ |
|
|
? |
$C_3.\He_3$ (as 27T20) |
$[3]$ |
$[6, 2]$ |
$2$ |
$26$ |
$69535904668008080000000$ |
$25$ |
$37$ |
| 27.27.122...929.2 |
$x^{27} - 9 x^{26} - 342 x^{25} + 1824 x^{24} + 51885 x^{23} - 78219 x^{22} - 4137969 x^{21} - 6575121 x^{20} + 168077007 x^{19} + 721384245 x^{18} - 2643326190 x^{17} - 24260405238 x^{16} - 23640008628 x^{15} + 287412684480 x^{14} + 1120649008689 x^{13} + 543185115909 x^{12} - 6657740592264 x^{11} - 21176998604370 x^{10} - 27997932326439 x^{9} - 9811914234453 x^{8} + 20214514093293 x^{7} + 28369977395991 x^{6} + 11645274040173 x^{5} - 2188102477347 x^{4} - 2616470753736 x^{3} - 238577790348 x^{2} + 76102592472 x - 3224160064$ |
$27$ |
(27, 0) |
$3^{70}\cdot 19^{24}$ |
$2$ |
$236.40186491792298$ |
$236.40186491792298$ |
|
|
|
$C_3.\He_3$ (as 27T20) |
$[9]$ |
$[9]$ |
$2$ |
$26$ |
$1213637600394272600000000$ |
$27$ |
$19$ |
| 27.27.122...929.7 |
$x^{27} - 342 x^{25} - 513 x^{24} + 39501 x^{23} + 103626 x^{22} - 2110482 x^{21} - 7594965 x^{20} + 56828943 x^{19} + 266968335 x^{18} - 743189256 x^{17} - 4964579217 x^{16} + 3264933096 x^{15} + 51511218516 x^{14} + 25933433526 x^{13} - 299879051373 x^{12} - 388790529075 x^{11} + 922034428029 x^{10} + 1946183510577 x^{9} - 1158915570840 x^{8} - 4748346050760 x^{7} - 525475322271 x^{6} + 5867137397565 x^{5} + 2747350748124 x^{4} - 3466946545884 x^{3} - 2385798454224 x^{2} + 762915465279 x + 655074933805$ |
$27$ |
(27, 0) |
$3^{70}\cdot 19^{24}$ |
$2$ |
$236.40186491792298$ |
$236.40186491792298$ |
|
|
? |
$C_3.\He_3$ (as 27T20) |
$[9]$ |
$[9]$ |
$2$ |
$26$ |
$89028481592346180000000$ |
$27$ |
$19$ |
| 27.27.108...889.3 |
$x^{27} - 666 x^{25} - 2997 x^{24} + 146853 x^{23} + 1147185 x^{22} - 12715272 x^{21} - 149019831 x^{20} + 327781890 x^{19} + 8491743090 x^{18} + 9647232111 x^{17} - 237954006801 x^{16} - 712854778653 x^{15} + 3293046015642 x^{14} + 15681239636427 x^{13} - 18808874095536 x^{12} - 165124042326342 x^{11} - 14147471336031 x^{10} + 897820958000655 x^{9} + 630160705915587 x^{8} - 2412816711352704 x^{7} - 2587671511709517 x^{6} + 2531891815664094 x^{5} + 3938355494393814 x^{4} + 97061946865029 x^{3} - 1686304839652515 x^{2} - 789687143156223 x - 108414539194410$ |
$27$ |
(27, 0) |
$3^{70}\cdot 37^{24}$ |
$2$ |
$427.5019684687372$ |
$427.5019684687372$ |
|
|
|
$C_3.\He_3$ (as 27T20) |
$[9]$ |
$[18, 2, 2, 2, 2, 2, 2, 2]$ |
$2$ |
$26$ |
$6040830012644733000000000000$ |
$19$ |
$37$ |
| 27.27.108...889.8 |
$x^{27} - 783 x^{25} - 3735 x^{24} + 234522 x^{23} + 2155005 x^{22} - 29276343 x^{21} - 450464463 x^{20} + 571622886 x^{19} + 38900814921 x^{18} + 161918673246 x^{17} - 1031632319232 x^{16} - 9916924254003 x^{15} - 9707964737067 x^{14} + 160519787099298 x^{13} + 542886228493821 x^{12} - 741667921495578 x^{11} - 6332226097217526 x^{10} - 4184870936717556 x^{9} + 31007228138488998 x^{8} + 54458037244128699 x^{7} - 52093139506177410 x^{6} - 190229982854248899 x^{5} - 61208132195516346 x^{4} + 203692811689213767 x^{3} + 210076438036406886 x^{2} + 64868673272521761 x + 5135556028428063$ |
$27$ |
(27, 0) |
$3^{70}\cdot 37^{24}$ |
$2$ |
$427.5019684687372$ |
$427.5019684687372$ |
|
|
? |
$C_3.\He_3$ (as 27T20) |
$[9]$ |
$[18, 2]$ |
$2$ |
$26$ |
$196043358370465300000000000$ |
$25$ |
$37$ |
