| Label |
$n$ |
Polynomial |
$p$ |
$f$ |
$e$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible Artin slopes |
Visible Swan slopes |
Artin slope content |
Swan slope content |
Hidden Artin slopes |
Hidden Swan slopes |
$\#\Aut(K/\Q_p)$ |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
Resid. Poly |
Jump Set |
| 2.1.8.10b1.1 |
$8$ |
$x^{8} + 2 x^{3} + 2 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$10$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]$ |
$[\frac{1}{3},\frac{1}{3},\frac{1}{2}]$ |
$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},\frac{1}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 2 x^{3} + 2 x^{2} + 2$ |
$[3, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 11]$ |
| 2.1.8.10b1.2 |
$8$ |
$x^{8} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$10$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]$ |
$[\frac{1}{3},\frac{1}{3},\frac{1}{2}]$ |
$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},\frac{1}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2$ |
$[3, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 11]$ |
| 2.1.8.12b1.1 |
$8$ |
$x^{8} + 2 x^{5} + 2 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$12$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, 2]$ |
$[\frac{1}{3},\frac{1}{3},1]$ |
$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 2 x^{5} + 2 x^{2} + 2$ |
$[5, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 15]$ |
| 2.1.8.12b1.2 |
$8$ |
$x^{8} + 2 x^{5} + 2 x^{2} + 6$ |
$2$ |
$1$ |
$8$ |
$12$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, 2]$ |
$[\frac{1}{3},\frac{1}{3},1]$ |
$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 2 x^{5} + 2 x^{2} + 6$ |
$[5, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 15]$ |
| 2.1.8.16b1.3 |
$8$ |
$x^{8} + 4 x^{7} + 2 x^{2} + 4 x + 2$ |
$2$ |
$1$ |
$8$ |
$16$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, 3]$ |
$[\frac{1}{3},\frac{1}{3},2]$ |
$[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 2 x^{2} + 4 x + 2$ |
$[9, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 13]$ |
| 2.1.8.16b1.4 |
$8$ |
$x^{8} + 4 x^{7} + 2 x^{2} + 4 x + 10$ |
$2$ |
$1$ |
$8$ |
$16$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, 3]$ |
$[\frac{1}{3},\frac{1}{3},2]$ |
$[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 2 x^{2} + 4 x + 10$ |
$[9, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 13]$ |
| 2.1.8.16b1.5 |
$8$ |
$x^{8} + 4 x^{5} + 2 x^{2} + 4 x + 2$ |
$2$ |
$1$ |
$8$ |
$16$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, 3]$ |
$[\frac{1}{3},\frac{1}{3},2]$ |
$[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{5} + 2 x^{2} + 4 x + 2$ |
$[9, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 13]$ |
| 2.1.8.16b1.6 |
$8$ |
$x^{8} + 4 x^{5} + 2 x^{2} + 4 x + 10$ |
$2$ |
$1$ |
$8$ |
$16$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{4}{3}, \frac{4}{3}, 3]$ |
$[\frac{1}{3},\frac{1}{3},2]$ |
$[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{5} + 2 x^{2} + 4 x + 10$ |
$[9, 2, 2, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 2, 5, 13]$ |
| 2.1.8.22a1.1 |
$8$ |
$x^{8} + 4 x^{7} + 4 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 4 x^{2} + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.2 |
$8$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 4 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 4 x^{2} + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.3 |
$8$ |
$x^{8} + 4 x^{7} + 8 x^{3} + 4 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 8 x^{3} + 4 x^{2} + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.4 |
$8$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.5 |
$8$ |
$x^{8} + 4 x^{7} + 4 x^{2} + 8 x + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 4 x^{2} + 8 x + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.6 |
$8$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 4 x^{2} + 8 x + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 4 x^{2} + 8 x + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.7 |
$8$ |
$x^{8} + 4 x^{7} + 8 x^{3} + 4 x^{2} + 8 x + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 8 x^{3} + 4 x^{2} + 8 x + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22a1.8 |
$8$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$ |
$2$ |
$1$ |
$8$ |
$22$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
$[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},\frac{5}{2}]_{3}^{2}$ |
$[\ ]^{2}_{3}$ |
$[\ ]^{2}_{3}$ |
$2$ |
$t + 1$ |
$x^{8} + 4 x^{7} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$ |
$[15, 10, 8, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
