| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 2.8.0.1 |
$8$ |
$x^{8} + x^{4} + x^{3} + x^{2} + 1$ |
$2$ |
$1$ |
$8$ |
$0$ |
$C_8$ (as 8T1) |
$8$ |
$1$ |
$[\ ]$ |
$[\ ]^{8}$ |
$t^{8} + t^{4} + t^{3} + t^{2} + 1$ |
$x - 2$ |
$[0]$ |
$[]$ |
| 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) |
$4$ |
$1$ |
$[2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(2 t^{2} + 2\right) x + 6$ |
$[1, 0]$ |
$[1]$ |
| 2.8.10.1 |
$8$ |
$x^{8} + 2 x^{3} + 2$ |
$2$ |
$8$ |
$1$ |
$10$ |
$C_2^3:(C_7: C_3)$ (as 8T36) |
$3$ |
$7$ |
$[10/7, 10/7, 10/7]$ |
$[10/7, 10/7, 10/7]_{7}^{3}$ |
$t + 1$ |
$x^{8} + 2 x^{3} + 2$ |
$[3, 3, 3, 0]$ |
$[1]$ |
| 2.8.10.2 |
$8$ |
$x^{8} + 2 x^{3} + 2 x^{2} + 2$ |
$2$ |
$8$ |
$1$ |
$10$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[4/3, 4/3, 3/2]$ |
$[4/3, 4/3, 3/2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{3} + 2 x^{2} + 2$ |
$[3, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.10.3 |
$8$ |
$x^{8} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2$ |
$2$ |
$8$ |
$1$ |
$10$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[4/3, 4/3, 3/2]$ |
$[4/3, 4/3, 3/2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2$ |
$[3, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.12.1 |
$8$ |
$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_4\times C_2$ (as 8T2) |
$4$ |
$1$ |
$[3]$ |
$[3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t^{3} + 4 t^{2}\right) x + 12 t^{3} + 12 t^{2} + 4 t + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.10 |
$8$ |
$x^{8} + 68 x^{6} + 272 x^{5} + 928 x^{4} + 5376 x^{3} + 13680 x^{2} + 15552 x + 92016$ |
$2$ |
$2$ |
$4$ |
$12$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[3]$ |
$[2, 2, 2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t^{2} + 4 t\right) x + 12 t^{3} + 12 t^{2} + 6$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.11 |
$8$ |
$x^{8} + 16 x^{7} + 92 x^{6} + 352 x^{5} + 1264 x^{4} + 2304 x^{3} + 6544 x^{2} - 256 x + 12080$ |
$2$ |
$2$ |
$4$ |
$12$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[3]$ |
$[2, 2, 2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t + 4\right) x + 4 t^{3} + 12 t^{2} + 4 t + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.12 |
$8$ |
$x^{8} + 92 x^{6} + 144 x^{5} + 1232 x^{4} + 5248 x^{3} + 11152 x^{2} + 34752 x + 49072$ |
$2$ |
$2$ |
$4$ |
$12$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[3]$ |
$[2, 2, 2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t^{2} + 4 t\right) x + 4 t^{3} + 12 t^{2} + 6$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.13 |
$8$ |
$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$ |
$2$ |
$4$ |
$2$ |
$12$ |
$D_4$ (as 8T4) |
$2$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{2}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{3} + 2$ |
$[3, 3, 0]$ |
$[1]$ |
| 2.8.12.14 |
$8$ |
$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$ |
$2$ |
$4$ |
$2$ |
$12$ |
$D_4$ (as 8T4) |
$2$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{2}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{3} + 6$ |
$[3, 3, 0]$ |
$[1]$ |
| 2.8.12.15 |
$8$ |
$x^{8} + 4 x^{7} + 4 x^{6} + 12$ |
$2$ |
$4$ |
$2$ |
$12$ |
$C_2^2:C_4$ (as 8T10) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{3} + 4 t + 2$ |
$[3, 3, 0]$ |
$[1]$ |
| 2.8.12.16 |
$8$ |
$x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$ |
$2$ |
$4$ |
$2$ |
$12$ |
$A_4\times C_2$ (as 8T13) |
$6$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{6}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{3} + 2 x^{2} + 6$ |
$[3, 2, 0]$ |
$[3]$ |
| 2.8.12.17 |
$8$ |
$x^{8} + 4 x^{7} + 6 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 4$ |
$2$ |
$4$ |
$2$ |
$12$ |
$C_2^4:C_6$ (as 8T33) |
$6$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2, 2]^{6}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 x^{3} + \left(2 t + 2\right) x^{2} + 2$ |
$[3, 2, 0]$ |
$[3]$ |
| 2.8.12.18 |
$8$ |
$x^{8} - 2 x^{7} + 4 x^{6} + 8 x^{4} + 4 x^{3} + 28$ |
$2$ |
$4$ |
$2$ |
$12$ |
$C_2^4:C_6$ (as 8T33) |
$6$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2, 2]^{6}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 t x^{3} + 4 t + 6$ |
$[3, 3, 0]$ |
$[3]$ |
| 2.8.12.19 |
$8$ |
$x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ |
$2$ |
$4$ |
$2$ |
$12$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 t x^{3} + 2 t x^{2} + 4 t + 2$ |
$[3, 2, 0]$ |
$[2]$ |
| 2.8.12.2 |
$8$ |
$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_4\times C_2$ (as 8T2) |
$4$ |
$1$ |
$[3]$ |
$[3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 4 x + 6$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.20 |
$8$ |
$x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 8 x^{4} - 4 x^{3} - 4 x^{2} + 4$ |
$2$ |
$4$ |
$2$ |
$12$ |
$C_2^3: C_4$ (as 8T21) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 t x^{3} + 2 t x^{2} + 2$ |
$[3, 2, 0]$ |
$[2]$ |
| 2.8.12.21 |
$8$ |
$x^{8} + 2 x^{7} + 2 x^{6} + 4 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 4$ |
$2$ |
$4$ |
$2$ |
$12$ |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + \left(2 t + 2\right) x^{3} + 2 t x^{2} + 2$ |
$[3, 2, 0]$ |
$[2]$ |
| 2.8.12.22 |
$8$ |
$x^{8} + 2 x^{7} + 8 x^{6} + 4 x^{5} + 4 x^{4} + 12 x^{3} + 12$ |
$2$ |
$4$ |
$2$ |
$12$ |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + \left(2 t + 2\right) x^{3} + 2 x^{2} + 4 t + 2$ |
$[3, 2, 0]$ |
$[2]$ |
| 2.8.12.23 |
$8$ |
$x^{8} - 2 x^{7} + 8 x^{6} - 4 x^{5} + 8 x^{4} - 4 x^{3} + 8 x^{2} + 4$ |
$2$ |
$4$ |
$2$ |
$12$ |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + 2 t x^{3} + 2 x^{2} + 2$ |
$[3, 2, 0]$ |
$[2]$ |
| 2.8.12.24 |
$8$ |
$x^{8} + 2 x^{7} + 2 x^{6} + 4 x^{5} + 16 x^{4} + 12 x^{3} - 12 x^{2} + 36$ |
$2$ |
$4$ |
$2$ |
$12$ |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) |
$4$ |
$1$ |
$[2, 2]$ |
$[2, 2, 2, 2]^{4}$ |
$t^{2} + t + 1$ |
$x^{4} + \left(2 t + 2\right) x^{3} + 2 t x^{2} + 6$ |
$[3, 2, 0]$ |
$[2]$ |
| 2.8.12.25 |
$8$ |
$x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ |
$2$ |
$8$ |
$1$ |
$12$ |
$S_4\times C_2$ (as 8T24) |
$2$ |
$3$ |
$[4/3, 4/3, 2]$ |
$[4/3, 4/3, 2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ |
$[5, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.12.26 |
$8$ |
$x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 6$ |
$2$ |
$8$ |
$1$ |
$12$ |
$S_4\times C_2$ (as 8T24) |
$2$ |
$3$ |
$[4/3, 4/3, 2]$ |
$[4/3, 4/3, 2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 6$ |
$[5, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.12.27 |
$8$ |
$x^{8} + 2 x^{5} + 2 x^{4} + 2$ |
$2$ |
$8$ |
$1$ |
$12$ |
$C_2^3:(C_7: C_3)$ (as 8T36) |
$3$ |
$7$ |
$[12/7, 12/7, 12/7]$ |
$[12/7, 12/7, 12/7]_{7}^{3}$ |
$t + 1$ |
$x^{8} + 2 x^{5} + 2 x^{4} + 2$ |
$[5, 5, 4, 0]$ |
$[1]$ |
| 2.8.12.28 |
$8$ |
$x^{8} + 2 x^{5} + 2$ |
$2$ |
$8$ |
$1$ |
$12$ |
$C_2^3:(C_7: C_3)$ (as 8T36) |
$3$ |
$7$ |
$[12/7, 12/7, 12/7]$ |
$[12/7, 12/7, 12/7]_{7}^{3}$ |
$t + 1$ |
$x^{8} + 2 x^{5} + 2$ |
$[5, 5, 5, 0]$ |
$[1]$ |
| 2.8.12.29 |
$8$ |
$x^{8} + 2 x^{5} + 2 x^{2} + 6$ |
$2$ |
$8$ |
$1$ |
$12$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[4/3, 4/3, 2]$ |
$[4/3, 4/3, 2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{5} + 2 x^{2} + 6$ |
$[5, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.12.3 |
$8$ |
$x^{8} - 12 x^{7} + 44 x^{6} + 152 x^{5} - 512 x^{4} + 560 x^{3} + 4432 x^{2} + 9376 x + 21744$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_2^2:C_4$ (as 8T10) |
$4$ |
$1$ |
$[3]$ |
$[2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 4 t^{3} x + 4 t^{3} + 12 t + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.30 |
$8$ |
$x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ |
$2$ |
$8$ |
$1$ |
$12$ |
$\textrm{GL(2,3)}$ (as 8T23) |
$2$ |
$3$ |
$[4/3, 4/3, 2]$ |
$[4/3, 4/3, 2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ |
$[5, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.12.4 |
$8$ |
$x^{8} + 8 x^{6} - 16 x^{5} - 376 x^{4} - 128 x^{3} + 8928 x^{2} + 21056 x + 13008$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_2^3: C_4$ (as 8T20) |
$4$ |
$1$ |
$[3]$ |
$[2, 2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 4 t x + 4 t^{2} + 12 t + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.5 |
$8$ |
$x^{8} + 208 x^{5} - 88 x^{4} - 1344 x^{3} + 13440 x^{2} - 16320 x + 17744$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_8$ (as 8T1) |
$4$ |
$1$ |
$[3]$ |
$[3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 4 t x + 8 t^{3} + 12 t^{2} + 8 t + 6$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.6 |
$8$ |
$x^{8} - 16 x^{6} + 72 x^{4} + 3664$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_8$ (as 8T1) |
$4$ |
$1$ |
$[3]$ |
$[3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 8 t^{3} + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.7 |
$8$ |
$x^{8} + 16 x^{7} + 160 x^{6} + 800 x^{5} + 2952 x^{4} + 7104 x^{3} + 12992 x^{2} + 14208 x + 10320$ |
$2$ |
$2$ |
$4$ |
$12$ |
$C_8:C_2$ (as 8T7) |
$4$ |
$1$ |
$[3]$ |
$[2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t^{2} + 4 t + 4\right) x + 8 t^{3} + 8 t^{2} + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.8 |
$8$ |
$x^{8} - 16 x^{6} + 112 x^{5} + 56 x^{4} - 1280 x^{3} + 10176 x^{2} + 39744 x + 40720$ |
$2$ |
$2$ |
$4$ |
$12$ |
$(C_8:C_2):C_2$ (as 8T16) |
$4$ |
$1$ |
$[3]$ |
$[2, 2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + 4 t x + 8 t^{3} + 4 t^{2} + 12 t + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.12.9 |
$8$ |
$x^{8} + 16 x^{7} + 68 x^{6} + 192 x^{5} + 1504 x^{4} + 2560 x^{3} + 12016 x^{2} + 13184 x + 15216$ |
$2$ |
$2$ |
$4$ |
$12$ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$4$ |
$1$ |
$[3]$ |
$[2, 2, 2, 3]^{4}$ |
$t^{4} + t + 1$ |
$x^{2} + \left(4 t + 4\right) x + 12 t^{3} + 12 t^{2} + 4 t + 2$ |
$[2, 0]$ |
$[1]$ |
| 2.8.14.1 |
$8$ |
$x^{8} + 2 x^{7} + 6$ |
$2$ |
$8$ |
$1$ |
$14$ |
$A_4\times C_2$ (as 8T13) |
$3$ |
$1$ |
$[2, 2, 2]$ |
$[2, 2, 2]^{3}$ |
$t + 1$ |
$x^{8} + 2 x^{7} + 6$ |
$[7, 7, 7, 0]$ |
$[3]$ |
| 2.8.14.10 |
$8$ |
$x^{8} + 2 x^{7} + 2 x^{2} + 4 x + 2$ |
$2$ |
$8$ |
$1$ |
$14$ |
$C_2 \wr S_4$ (as 8T44) |
$2$ |
$3$ |
$[4/3, 4/3, 5/2]$ |
$[4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{7} + 2 x^{2} + 4 x + 2$ |
$[7, 2, 2, 0]$ |
$[1, 1]$ |
| 2.8.14.11 |
$8$ |
$x^{8} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 2$ |
$2$ |
$8$ |
$1$ |
$14$ |
$C_2 \wr S_4$ (as 8T44) |
$2$ |
$3$ |
$[4/3, 4/3, 5/2]$ |
$[4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$ |
$t + 1$ |
$x^{8} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 2$ |
$[7, 2, 2, 0]$ |
$[1, 1]$ |
