| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 3.6.10.1 |
$6$ |
$x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90$ |
$3$ |
$3$ |
$2$ |
$10$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(9 t + 18\right) x + 9 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.6.10.10 |
$6$ |
$x^{6} + 3 x^{5} + 18 x^{2} + 18 x + 15$ |
$3$ |
$6$ |
$1$ |
$10$ |
$C_3^2:D_4$ (as 6T13) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t + 1$ |
$x^{6} + 3 x^{5} + 18 x^{2} + 18 x + 15$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.6.10.11 |
$6$ |
$x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$ |
$3$ |
$6$ |
$1$ |
$10$ |
$C_3^2:D_4$ (as 6T13) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t + 1$ |
$x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.6.10.12 |
$6$ |
$x^{6} + 3 x^{5} + 3$ |
$3$ |
$6$ |
$1$ |
$10$ |
$C_3^2:D_4$ (as 6T13) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t + 1$ |
$x^{6} + 3 x^{5} + 3$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.6.10.2 |
$6$ |
$x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$ |
$3$ |
$3$ |
$2$ |
$10$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 18 t x + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.6.10.3 |
$6$ |
$x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ |
$3$ |
$3$ |
$2$ |
$10$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 9 x + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.6.10.4 |
$6$ |
$x^{6} + 6 x^{3} + 81 x^{2} + 9$ |
$3$ |
$3$ |
$2$ |
$10$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(9 t + 9\right) x + 3$ |
$[3, 0]$ |
$[1]$ |
| 3.6.10.5 |
$6$ |
$x^{6} - 12 x^{3} + 360$ |
$3$ |
$3$ |
$2$ |
$10$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.6.10.6 |
$6$ |
$x^{6} - 18 x^{4} - 12 x^{3} + 162 x^{2} + 432 x + 360$ |
$3$ |
$3$ |
$2$ |
$10$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 9 t x + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |
| 3.6.10.7 |
$6$ |
$x^{6} + 3 x^{5} + 6$ |
$3$ |
$6$ |
$1$ |
$10$ |
$C_3^2:D_4$ (as 6T13) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t + 1$ |
$x^{6} + 3 x^{5} + 6$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.6.10.8 |
$6$ |
$x^{6} + 3 x^{5} + 18 x^{2} + 18 x + 21$ |
$3$ |
$6$ |
$1$ |
$10$ |
$C_3^2:D_4$ (as 6T13) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t + 1$ |
$x^{6} + 3 x^{5} + 18 x^{2} + 18 x + 21$ |
$[5, 0]$ |
$[1, 1]$ |
| 3.6.10.9 |
$6$ |
$x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 12$ |
$3$ |
$6$ |
$1$ |
$10$ |
$C_3^2:D_4$ (as 6T13) |
$2$ |
$4$ |
$[9/4]$ |
$[9/4, 9/4]_{4}^{2}$ |
$t + 1$ |
$x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 12$ |
$[5, 0]$ |
$[1, 1]$ |
