| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 7.8.0.1 |
$8$ |
$x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3$ |
$7$ |
$1$ |
$8$ |
$0$ |
$C_8$ (as 8T1) |
$8$ |
$1$ |
$[\ ]$ |
$[\ ]^{8}$ |
$t^{8} + 4 t^{3} + 6 t^{2} + 2 t + 3$ |
$x - 7$ |
$[0]$ |
$[]$ |
| 7.8.4.1 |
$8$ |
$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ |
$7$ |
$2$ |
$4$ |
$4$ |
$C_4\times C_2$ (as 8T2) |
$4$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{4}$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{2} + 7$ |
$[0]$ |
$[1]$ |
| 7.8.4.2 |
$8$ |
$x^{8} + 245 x^{4} - 1372 x^{2} + 7203$ |
$7$ |
$2$ |
$4$ |
$4$ |
$C_8$ (as 8T1) |
$4$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{4}$ |
$t^{4} + 5 t^{2} + 4 t + 3$ |
$x^{2} + 7 t$ |
$[0]$ |
$[1]$ |
| 7.8.6.1 |
$8$ |
$x^{8} + 14 x^{4} - 245$ |
$7$ |
$4$ |
$2$ |
$6$ |
$Q_8$ (as 8T5) |
$2$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{4} + 7 t + 28$ |
$[0]$ |
$[1]$ |
| 7.8.6.2 |
$8$ |
$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ |
$7$ |
$4$ |
$2$ |
$6$ |
$D_4$ (as 8T4) |
$2$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{2}$ |
$t^{2} + 6 t + 3$ |
$x^{4} + 7$ |
$[0]$ |
$[1]$ |
| 7.8.6.3 |
$8$ |
$x^{8} - 154 x^{4} - 1421$ |
$7$ |
$4$ |
$2$ |
$6$ |
$C_8:C_2$ (as 8T7) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$t^{2} + 6 t + 3$ |
$x^{4} + 35 t + 28$ |
$[0]$ |
$[1]$ |
| 7.8.7.1 |
$8$ |
$x^{8} + 7$ |
$7$ |
$8$ |
$1$ |
$7$ |
$D_{8}$ (as 8T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$t + 4$ |
$x^{8} + 7$ |
$[0]$ |
$[2]$ |
| 7.8.7.2 |
$8$ |
$x^{8} + 21$ |
$7$ |
$8$ |
$1$ |
$7$ |
$D_{8}$ (as 8T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$t + 4$ |
$x^{8} + 21$ |
$[0]$ |
$[2]$ |
