| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 3.12.12.23 |
$12$ |
$x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ |
$3$ |
$3$ |
$4$ |
$12$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(3 t + 3\right) x^{2} + 3 x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.12.12.5 |
$12$ |
$x^{12} - 12 x^{11} + 132 x^{10} - 204 x^{9} + 3024 x^{8} - 432 x^{7} + 3510 x^{6} + 2268 x^{5} - 972 x^{4} + 756 x^{3} + 1620 x^{2} + 648 x + 81$ |
$3$ |
$3$ |
$4$ |
$12$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(6 t^{3} + 3 t^{2} + 6\right) x^{2} + 6 x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.12.14.3 |
$12$ |
$x^{12} + 30 x^{6} + 72 x^{5} + 36 x^{4} + 36 x^{3} + 36 x^{2} + 18$ |
$3$ |
$6$ |
$2$ |
$14$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + \left(6 t + 6\right) x^{3} + \left(6 t + 6\right) x^{2} + 3 t$ |
$[2, 0]$ |
$[2, 1]$ |
| 3.12.14.9 |
$12$ |
$x^{12} - 6 x^{9} + 12 x^{6} + 18 x^{5} + 9 x^{4} + 36 x^{3} + 18 x^{2} + 18$ |
$3$ |
$6$ |
$2$ |
$14$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 t x^{3} + \left(3 t + 3\right) x^{2} + 3 t$ |
$[2, 0]$ |
$[2, 1]$ |
| 3.12.20.20 |
$12$ |
$x^{12} - 36 x^{11} + 72 x^{10} - 1896 x^{9} + 210600 x^{8} - 32400 x^{7} - 22464 x^{6} - 178848 x^{5} - 283824 x^{4} + 6912 x^{3} + 217728 x^{2} + 124416 x + 20736$ |
$3$ |
$3$ |
$4$ |
$20$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + 18 t x^{2} + 18 x + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.23 |
$12$ |
$x^{12} - 90 x^{11} + 3096 x^{10} - 26718 x^{9} + 681291 x^{8} + 2423196 x^{7} + 5699376 x^{6} + 7319646 x^{5} + 6384096 x^{4} + 6514452 x^{3} + 5675022 x^{2} + 2439720 x + 411237$ |
$3$ |
$3$ |
$4$ |
$20$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + \left(18 t^{3} + 9 t^{2} + 9 t + 9\right) x^{2} + 18 t^{3} x + 9 t^{3} + 9 t^{2} + 18 t + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.20.5 |
$12$ |
$x^{12} - 144 x^{11} + 7668 x^{10} - 129660 x^{9} + 708264 x^{8} + 912708 x^{7} + 1567890 x^{6} + 7402752 x^{5} + 19458468 x^{4} + 27085860 x^{3} + 22233528 x^{2} + 10477188 x + 2365929$ |
$3$ |
$3$ |
$4$ |
$20$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{4} + 2 t^{3} + 2$ |
$x^{3} + 18 t^{3} x^{2} + \left(18 t^{3} + 18 t + 18\right) x + 18 t^{3} + 9 t^{2} + 18 t + 21$ |
$[3, 0]$ |
$[1]$ |
| 3.12.22.12 |
$12$ |
$x^{12} + 30 x^{9} - 36 x^{8} + 219 x^{6} - 540 x^{5} + 648 x^{4} + 234 x^{3} + 945 x^{2} + 378 x + 450$ |
$3$ |
$6$ |
$2$ |
$22$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 15 x^{3} + 18 t x^{2} + \left(9 t + 9\right) x + 21 t + 18$ |
$[6, 0]$ |
$[2, 1]$ |
| 3.12.22.26 |
$12$ |
$x^{12} + 6 x^{9} - 15 x^{6} + 324 x^{4} + 576 x^{3} + 1080 x^{2} + 756 x + 585$ |
$3$ |
$6$ |
$2$ |
$22$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 x^{3} + \left(18 t + 18\right) x^{2} + \left(18 t + 18\right) x + 21 t + 9$ |
$[6, 0]$ |
$[2, 1]$ |
| 3.12.22.41 |
$12$ |
$x^{12} - 6 x^{9} - 18 x^{7} + 30 x^{6} + 108 x^{4} - 18 x^{3} + 162 x^{2} - 54 x + 45$ |
$3$ |
$6$ |
$2$ |
$22$ |
$S_3 \times C_4$ (as 12T11) |
$4$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{6} + 3 t x^{3} + 9 t x + 3 t + 9$ |
$[6, 0]$ |
$[2, 1]$ |
