| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 2.8.8.1 |
$8$ |
$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ |
$2$ |
$2$ |
$4$ |
$8$ |
$C_4\times C_2$ (as 8T2) |
$4$ |
$1$ |
$[2]$ |
$[2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 2 x + 2$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.10 |
$8$ |
$x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ |
$2$ |
$2$ |
$4$ |
$8$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{3} + 2 t\right) x + 4 t + 2$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.11 |
$8$ |
$x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ |
$2$ |
$4$ |
$2$ |
$8$ |
$S_4$ (as 8T14) |
$2$ |
$3$ |
$[4/3, 4/3]$ |
$[4/3, 4/3]_{3}^{2}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{2} + 2 x + 2$ |
$[1, 1, 0]$ |
$[1]$ |
| 2.8.8.12 |
$8$ |
$x^{8} - 2 x^{6} - 2 x^{5} + 8 x^{4} + 8 x^{3} - 4 x + 4$ |
$2$ |
$4$ |
$2$ |
$8$ |
$A_4\wr C_2$ (as 8T42) |
$6$ |
$3$ |
$[4/3, 4/3]$ |
$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 t x^{2} + 2 t x + 2$ |
$[1, 1, 0]$ |
$[1]$ |
| 2.8.8.13 |
$8$ |
$x^{8} + 2 x + 2$ |
$2$ |
$8$ |
$1$ |
$8$ |
$C_2^3:(C_7: C_3)$ (as 8T36) |
$3$ |
$7$ |
$[8/7, 8/7, 8/7]$ |
$[8/7, 8/7, 8/7]_{7}^{3}$ |
$t + 1$ |
$x^{8} + 2 x + 2$ |
$[1, 1, 1, 0]$ |
$[1]$ |
| 2.8.8.2 |
$8$ |
$x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ |
$2$ |
$2$ |
$4$ |
$8$ |
$C_2^2:C_4$ (as 8T10) |
$4$ |
$1$ |
$[2]$ |
$[2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 4 t^{3} + 4 t^{2} + 6$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.3 |
$8$ |
$x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ |
$2$ |
$2$ |
$4$ |
$8$ |
$C_2^3: C_4$ (as 8T20) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{3} + 2 t^{2} + 2\right) x + 6$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.4 |
$8$ |
$x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ |
$2$ |
$2$ |
$4$ |
$8$ |
$C_8$ (as 8T1) |
$4$ |
$1$ |
$[2]$ |
$[2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 2 x + 4 t^{3} + 4 t + 2$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.5 |
$8$ |
$x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$ |
$2$ |
$2$ |
$4$ |
$8$ |
$C_8:C_2$ (as 8T7) |
$4$ |
$1$ |
$[2]$ |
$[2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 4 t^{3} + 6$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.6 |
$8$ |
$x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ |
$2$ |
$2$ |
$4$ |
$8$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{3} + 2 t + 2\right) x + 4 t^{2} + 4 t + 2$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.7 |
$8$ |
$x^{8} + 8 x^{7} + 40 x^{6} + 120 x^{5} + 232 x^{4} + 240 x^{3} + 160 x^{2} + 64 x + 16$ |
$2$ |
$2$ |
$4$ |
$8$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{2} + 2\right) x + 2$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.8 |
$8$ |
$x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ |
$2$ |
$2$ |
$4$ |
$8$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{3} + 2 t\right) x + 4 t^{3} + 4 t + 6$ |
$[1, 0]$ |
$[1]$ |
| 2.8.8.9 |
$8$ |
$x^{8} + 8 x^{7} + 56 x^{6} + 216 x^{5} + 680 x^{4} + 1296 x^{3} + 2016 x^{2} + 1728 x + 1296$ |
$2$ |
$2$ |
$4$ |
$8$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{2} + 2\right) x + 6$ |
$[1, 0]$ |
$[1]$ |
