| 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 |
| 7.8.1.0a1.1 |
$8$ |
$x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3$ |
$7$ |
$8$ |
$1$ |
$0$ |
$C_8$ (as 8T1) |
$8$ |
$1$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]^{8}$ |
$[\ ]^{8}$ |
$[\ ]$ |
$[\ ]$ |
$8$ |
$t^{8} + 4 t^{3} + 6 t^{2} + 2 t + 3$ |
$x - 7$ |
$[0]$ |
$[\ ]$ |
|
undefined |
| 7.4.2.4a1.1 |
$8$ |
$( x^{4} + 5 x^{2} + 4 x + 3 )^{2} + 7 x$ |
$7$ |
$4$ |
$2$ |
$4$ |
$C_8$ (as 8T1) |
$4$ |
$2$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{2}^{4}$ |
$[\ ]_{2}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$8$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{2} + 7 t$ |
$[0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 7.4.2.4a1.2 |
$8$ |
$( x^{4} + 5 x^{2} + 4 x + 3 )^{2} + 7$ |
$7$ |
$4$ |
$2$ |
$4$ |
$C_4\times C_2$ (as 8T2) |
$4$ |
$2$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{2}^{4}$ |
$[\ ]_{2}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$8$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{2} + 7$ |
$[0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 7.2.4.6a1.1 |
$8$ |
$( x^{2} + 6 x + 3 )^{4} + 7 x$ |
$7$ |
$2$ |
$4$ |
$6$ |
$C_8:C_2$ (as 8T7) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$[\ ]_{4}^{4}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$4$ |
$t^{2} + 6 t + 3$ |
$x^{4} + 35 t + 28$ |
$[0]$ |
$[1]$ |
$z^{3} + 4 z^{2} + 6 z + 4$ |
undefined |
| 7.2.4.6a1.2 |
$8$ |
$( x^{2} + 6 x + 3 )^{4} + 7$ |
$7$ |
$2$ |
$4$ |
$6$ |
$D_4$ (as 8T4) |
$2$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{2}$ |
$[\ ]_{4}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$8$ |
$t^{2} + 6 t + 3$ |
$x^{4} + 7$ |
$[0]$ |
$[1]$ |
$z^{3} + 4 z^{2} + 6 z + 4$ |
undefined |
| 7.2.4.6a1.3 |
$8$ |
$( x^{2} + 6 x + 3 )^{4} + 7 x + 28$ |
$7$ |
$2$ |
$4$ |
$6$ |
$Q_8$ (as 8T5) |
$2$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{2}$ |
$[\ ]_{4}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$8$ |
$t^{2} + 6 t + 3$ |
$x^{4} + 7 t + 28$ |
$[0]$ |
$[1]$ |
$z^{3} + 4 z^{2} + 6 z + 4$ |
undefined |
| 7.1.8.7a1.1 |
$8$ |
$x^{8} + 7$ |
$7$ |
$1$ |
$8$ |
$7$ |
$D_{8}$ (as 8T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$[\ ]_{8}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$2$ |
$t + 4$ |
$x^{8} + 7$ |
$[0]$ |
$[2]$ |
$z^{7} + z^{6} + 1$ |
undefined |
| 7.1.8.7a1.2 |
$8$ |
$x^{8} + 21$ |
$7$ |
$1$ |
$8$ |
$7$ |
$D_{8}$ (as 8T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$[\ ]_{8}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$2$ |
$t + 4$ |
$x^{8} + 21$ |
$[0]$ |
$[2]$ |
$z^{7} + z^{6} + 1$ |
undefined |
