| 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 $p$ |
| 27.27.229...528.1 |
$x^{27} - 9 x^{26} - 39 x^{25} + 514 x^{24} + 327 x^{23} - 12030 x^{22} + 5923 x^{21} + 152622 x^{20} - 148550 x^{19} - 1166055 x^{18} + 1379370 x^{17} + 5630326 x^{16} - 6989285 x^{15} - 17537565 x^{14} + 21000324 x^{13} + 35322803 x^{12} - 37894050 x^{11} - 45471606 x^{10} + 39595704 x^{9} + 36423522 x^{8} - 21364000 x^{7} - 17146056 x^{6} + 4071159 x^{5} + 3996545 x^{4} + 311093 x^{3} - 146172 x^{2} - 15038 x + 881$ |
$27$ |
(27, 0) |
$2^{18}\cdot 3^{21}\cdot 7^{18}\cdot 431^{9}$ |
$4$ |
$103.11901510559684$ |
|
|
|
|
$C_3^3:S_3$ (as 27T46) |
trivial |
$[2, 2]$ |
$2$ |
$26$ |
$4384775773400145000$ |
$25$ |
$431$ |
| 27.27.261...608.1 |
$x^{27} - 78 x^{25} - 15 x^{24} + 2472 x^{23} + 831 x^{22} - 42152 x^{21} - 18525 x^{20} + 430203 x^{19} + 219679 x^{18} - 2762556 x^{17} - 1542348 x^{16} + 11392621 x^{15} + 6684192 x^{14} - 30155193 x^{13} - 18000857 x^{12} + 50276448 x^{11} + 29716908 x^{10} - 50817880 x^{9} - 29422410 x^{8} + 28955214 x^{7} + 16846006 x^{6} - 7959093 x^{5} - 5043156 x^{4} + 590784 x^{3} + 583725 x^{2} + 63735 x + 743$ |
$27$ |
(27, 0) |
$2^{12}\cdot 3^{27}\cdot 7^{18}\cdot 431^{9}$ |
$4$ |
$112.84183692090866$ |
|
|
|
|
$C_3^3:S_3$ (as 27T46) |
trivial |
$[2, 2, 2]$ |
$2$ |
$26$ |
$13044633162257166000$ |
$24$ |
$431$ |
| 27.27.167...912.1 |
$x^{27} - 78 x^{25} - 69 x^{24} + 2391 x^{23} + 3729 x^{22} - 36776 x^{21} - 78168 x^{20} + 303360 x^{19} + 833056 x^{18} - 1289328 x^{17} - 4920498 x^{16} + 1998961 x^{15} + 16364808 x^{14} + 3861165 x^{13} - 29580077 x^{12} - 18957114 x^{11} + 26856084 x^{10} + 25541057 x^{9} - 11892138 x^{8} - 16024680 x^{7} + 2411455 x^{6} + 5218554 x^{5} - 217893 x^{4} - 866850 x^{3} + 37389 x^{2} + 59427 x - 7561$ |
$27$ |
(27, 0) |
$2^{18}\cdot 3^{27}\cdot 7^{18}\cdot 431^{9}$ |
$4$ |
$131.63327964735237$ |
|
|
|
|
$C_3^3:S_3$ (as 27T46) |
$[2, 2]$ |
$[2, 2, 2, 2, 2]$ |
$2$ |
$26$ |
$41400174381407900000$ |
$24$ |
$431$ |
| 27.27.167...912.2 |
$x^{27} - 78 x^{25} - 3 x^{24} + 2571 x^{23} + 159 x^{22} - 47102 x^{21} - 3318 x^{20} + 531132 x^{19} + 35030 x^{18} - 3856650 x^{17} - 196776 x^{16} + 18340951 x^{15} + 528462 x^{14} - 57051909 x^{13} - 136513 x^{12} + 114046686 x^{11} - 2975622 x^{10} - 140501779 x^{9} + 7729512 x^{8} + 97589034 x^{7} - 8018767 x^{6} - 30948720 x^{5} + 3200613 x^{4} + 1972746 x^{3} - 245721 x^{2} - 19239 x + 2269$ |
$27$ |
(27, 0) |
$2^{18}\cdot 3^{27}\cdot 7^{18}\cdot 431^{9}$ |
$4$ |
$131.63327964735237$ |
|
|
|
|
$C_3^3:S_3$ (as 27T46) |
trivial |
$[2, 2, 2, 2, 2]$ |
$2$ |
$26$ |
$118615213928011760000$ |
$22$ |
$431$ |
