| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 5.12.0.1 |
$12$ |
$x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ |
$5$ |
$1$ |
$12$ |
$0$ |
$C_{12}$ (as 12T1) |
$12$ |
$1$ |
$[\ ]$ |
$[\ ]^{12}$ |
$t^{12} + t^{7} + t^{6} + 4 t^{4} + 4 t^{3} + 3 t^{2} + 2 t + 2$ |
$x - 5$ |
$[0]$ |
$[\ ]$ |
| 5.12.6.1 |
$12$ |
$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ |
$5$ |
$2$ |
$6$ |
$6$ |
$C_6\times C_2$ (as 12T2) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$t^{6} + t^{4} + 4 t^{3} + t^{2} + 2$ |
$x^{2} + 20 x + 5$ |
$[0]$ |
$[1]$ |
| 5.12.6.2 |
$12$ |
$x^{12} + 25 x^{8} - 500 x^{6} + 625 x^{4} + 31250$ |
$5$ |
$2$ |
$6$ |
$6$ |
$C_{12}$ (as 12T1) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$t^{6} + t^{4} + 4 t^{3} + t^{2} + 2$ |
$x^{2} + 5 t$ |
$[0]$ |
$[1]$ |
| 5.12.8.1 |
$12$ |
$x^{12} + 12 x^{10} + 32 x^{9} + 54 x^{8} + 96 x^{7} - 50 x^{6} + 240 x^{5} - 360 x^{4} - 884 x^{3} + 4044 x^{2} - 3912 x + 4173$ |
$5$ |
$3$ |
$4$ |
$8$ |
$C_3 : C_4$ (as 12T5) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{3} + 5$ |
$[0]$ |
$[1]$ |
| 5.12.8.2 |
$12$ |
$x^{12} + 100 x^{6} - 500 x^{3} + 1250$ |
$5$ |
$3$ |
$4$ |
$8$ |
$C_3\times (C_3 : C_4)$ (as 12T19) |
$12$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{12}$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{3} + 5 t$ |
$[0]$ |
$[1]$ |
| 5.12.9.1 |
$12$ |
$x^{12} - 30 x^{8} + 225 x^{4} + 1125$ |
$5$ |
$4$ |
$3$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 5 t^{2}$ |
$[0]$ |
$[1]$ |
| 5.12.9.2 |
$12$ |
$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ |
$5$ |
$4$ |
$3$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 5$ |
$[0]$ |
$[1]$ |
| 5.12.9.3 |
$12$ |
$x^{12} + 75 x^{4} - 375$ |
$5$ |
$4$ |
$3$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 5 t$ |
$[0]$ |
$[1]$ |
| 5.12.9.4 |
$12$ |
$x^{12} + 30 x^{8} + 600 x^{4} + 1000$ |
$5$ |
$4$ |
$3$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 10 t + 10$ |
$[0]$ |
$[1]$ |
| 5.12.10.1 |
$12$ |
$x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$ |
$5$ |
$6$ |
$2$ |
$10$ |
$D_6$ (as 12T3) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 4 t + 2$ |
$x^{6} + \left(5 t + 20\right) x + 5$ |
$[0]$ |
$[1]$ |
| 5.12.10.2 |
$12$ |
$x^{12} - 20 x^{6} - 100$ |
$5$ |
$6$ |
$2$ |
$10$ |
$C_6\times S_3$ (as 12T18) |
$6$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{6}$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 10 t + 10$ |
$[0]$ |
$[1]$ |
| 5.12.10.3 |
$12$ |
$x^{12} - 50 x^{6} - 175$ |
$5$ |
$6$ |
$2$ |
$10$ |
$C_3 : C_4$ (as 12T5) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 20 t + 15$ |
$[0]$ |
$[1]$ |
| 5.12.10.4 |
$12$ |
$x^{12} - 70 x^{6} + 425$ |
$5$ |
$6$ |
$2$ |
$10$ |
$C_3\times (C_3 : C_4)$ (as 12T19) |
$6$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{6}$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 20 t + 5$ |
$[0]$ |
$[1]$ |
| 5.12.11.1 |
$12$ |
$x^{12} + 20$ |
$5$ |
$12$ |
$1$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 3$ |
$x^{12} + 20$ |
$[0]$ |
$[2]$ |
| 5.12.11.2 |
$12$ |
$x^{12} + 5$ |
$5$ |
$12$ |
$1$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 3$ |
$x^{12} + 5$ |
$[0]$ |
$[2]$ |
| 5.12.11.3 |
$12$ |
$x^{12} + 10$ |
$5$ |
$12$ |
$1$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 3$ |
$x^{12} + 10$ |
$[0]$ |
$[2]$ |
| 5.12.11.4 |
$12$ |
$x^{12} + 15$ |
$5$ |
$12$ |
$1$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 3$ |
$x^{12} + 15$ |
$[0]$ |
$[2]$ |
