| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 7.12.0.1 |
$12$ |
$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$ |
$7$ |
$1$ |
$12$ |
$0$ |
$C_{12}$ (as 12T1) |
$12$ |
$1$ |
$[\ ]$ |
$[\ ]^{12}$ |
$t^{12} + 2 t^{8} + 5 t^{7} + 3 t^{6} + 2 t^{5} + 4 t^{4} + 5 t^{2} + 3$ |
$x - 7$ |
$[0]$ |
$[]$ |
| 7.12.6.1 |
$12$ |
$x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ |
$7$ |
$2$ |
$6$ |
$6$ |
$C_6\times C_2$ (as 12T2) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$t^{6} + t^{4} + 5 t^{3} + 4 t^{2} + 6 t + 3$ |
$x^{2} + 7$ |
$[0]$ |
$[1]$ |
| 7.12.6.2 |
$12$ |
$x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$ |
$7$ |
$2$ |
$6$ |
$6$ |
$C_{12}$ (as 12T1) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$t^{6} + t^{4} + 5 t^{3} + 4 t^{2} + 6 t + 3$ |
$x^{2} + 7 t$ |
$[0]$ |
$[1]$ |
| 7.12.8.1 |
$12$ |
$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ |
$7$ |
$3$ |
$4$ |
$8$ |
$C_{12}$ (as 12T1) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{3} + 7$ |
$[0]$ |
$[1]$ |
| 7.12.8.2 |
$12$ |
$x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$ |
$7$ |
$3$ |
$4$ |
$8$ |
$C_{12}$ (as 12T1) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{3} + 7 t^{2}$ |
$[0]$ |
$[1]$ |
| 7.12.8.3 |
$12$ |
$x^{12} + 245 x^{6} - 1372 x^{3} + 7203$ |
$7$ |
$3$ |
$4$ |
$8$ |
$C_{12}$ (as 12T1) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{3} + 7 t$ |
$[0]$ |
$[1]$ |
| 7.12.9.1 |
$12$ |
$x^{12} + 24 x^{11} + 216 x^{10} + 880 x^{9} + 1605 x^{8} + 2064 x^{7} + 6576 x^{6} + 11904 x^{5} + 8307 x^{4} - 50984 x^{3} - 57096 x^{2} + 58128 x + 76871$ |
$7$ |
$4$ |
$3$ |
$9$ |
$D_4 \times C_3$ (as 12T14) |
$6$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{6}$ |
$t^{3} + 6 t^{2} + 4$ |
$x^{4} + 7$ |
$[0]$ |
$[2]$ |
| 7.12.9.2 |
$12$ |
$x^{12} - 42 x^{8} - 1372$ |
$7$ |
$4$ |
$3$ |
$9$ |
$D_4 \times C_3$ (as 12T14) |
$6$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{6}$ |
$t^{3} + 6 t^{2} + 4$ |
$x^{4} + 7 t$ |
$[0]$ |
$[2]$ |
| 7.12.10.1 |
$12$ |
$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ |
$7$ |
$6$ |
$2$ |
$10$ |
$C_6\times C_2$ (as 12T2) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{6} + 7$ |
$[0]$ |
$[1]$ |
| 7.12.10.2 |
$12$ |
$x^{12} + 14 x^{6} - 245$ |
$7$ |
$6$ |
$2$ |
$10$ |
$C_6\times C_2$ (as 12T2) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{6} + 7 t + 28$ |
$[0]$ |
$[1]$ |
| 7.12.10.3 |
$12$ |
$x^{12} - 1176$ |
$7$ |
$6$ |
$2$ |
$10$ |
$C_6\times C_2$ (as 12T2) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{6} + 14 t + 42$ |
$[0]$ |
$[1]$ |
| 7.12.10.4 |
$12$ |
$x^{12} - 42 x^{6} + 147$ |
$7$ |
$6$ |
$2$ |
$10$ |
$C_{12}$ (as 12T1) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{6} + 7 t$ |
$[0]$ |
$[1]$ |
| 7.12.10.5 |
$12$ |
$x^{12} - 154 x^{6} - 1421$ |
$7$ |
$6$ |
$2$ |
$10$ |
$C_{12}$ (as 12T1) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{6} + 35 t + 28$ |
$[0]$ |
$[1]$ |
| 7.12.10.6 |
$12$ |
$x^{12} - 28 x^{6} - 98$ |
$7$ |
$6$ |
$2$ |
$10$ |
$C_{12}$ (as 12T1) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{6} + 7 t + 7$ |
$[0]$ |
$[1]$ |
| 7.12.11.1 |
$12$ |
$x^{12} + 14$ |
$7$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 4$ |
$x^{12} + 14$ |
$[0]$ |
$[2]$ |
| 7.12.11.2 |
$12$ |
$x^{12} + 7$ |
$7$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 4$ |
$x^{12} + 7$ |
$[0]$ |
$[2]$ |
| 7.12.11.3 |
$12$ |
$x^{12} + 28$ |
$7$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 4$ |
$x^{12} + 28$ |
$[0]$ |
$[2]$ |
| 7.12.11.4 |
$12$ |
$x^{12} + 42$ |
$7$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 4$ |
$x^{12} + 42$ |
$[0]$ |
$[2]$ |
| 7.12.11.5 |
$12$ |
$x^{12} + 21$ |
$7$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 4$ |
$x^{12} + 21$ |
$[0]$ |
$[2]$ |
| 7.12.11.6 |
$12$ |
$x^{12} + 35$ |
$7$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 4$ |
$x^{12} + 35$ |
$[0]$ |
$[2]$ |
