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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
2.8.0.1 $x^{8} + x^{4} + x^{3} + x^{2} + 1$ $2$ $1$ $8$ $0$ $C_8$ (as 8T1) $[\ ]$ $[\ ]^{8}$ $t^{8} + t^{4} + t^{3} + t^{2} + 1$ $x - 2$
2.8.8.1 $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ $2$ $2$ $4$ $8$ $C_4\times C_2$ (as 8T2) $[2]$ $[2]^{4}$ $t^{4} + t + 1$ $x^{2} + 2 x + 2$
2.8.8.2 $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ $2$ $2$ $4$ $8$ $C_2^2:C_4$ (as 8T10) $[2]$ $[2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 4 t^{3} + 4 t^{2} + 6$
2.8.8.3 $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ $2$ $2$ $4$ $8$ $C_2^3: C_4$ (as 8T20) $[2]$ $[2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t^{2} + 2\right) x + 6$
2.8.8.4 $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ $2$ $2$ $4$ $8$ $C_8$ (as 8T1) $[2]$ $[2]^{4}$ $t^{4} + t + 1$ $x^{2} + 2 x + 4 t^{3} + 4 t + 2$
2.8.8.5 $x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$ $2$ $2$ $4$ $8$ $C_8:C_2$ (as 8T7) $[2]$ $[2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 4 t^{3} + 6$
2.8.8.6 $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ $2$ $2$ $4$ $8$ $(C_8:C_2):C_2$ (as 8T16) $[2]$ $[2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t + 2\right) x + 4 t^{2} + 4 t + 2$
2.8.8.7 $x^{8} + 8 x^{7} + 40 x^{6} + 120 x^{5} + 232 x^{4} + 240 x^{3} + 160 x^{2} + 64 x + 16$ $2$ $2$ $4$ $8$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[2]$ $[2, 2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{2} + 2\right) x + 2$
2.8.8.8 $x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ $2$ $2$ $4$ $8$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[2]$ $[2, 2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t\right) x + 4 t^{3} + 4 t + 6$
2.8.8.9 $x^{8} + 8 x^{7} + 56 x^{6} + 216 x^{5} + 680 x^{4} + 1296 x^{3} + 2016 x^{2} + 1728 x + 1296$ $2$ $2$ $4$ $8$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[2]$ $[2, 2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{2} + 2\right) x + 6$
2.8.8.10 $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ $2$ $2$ $4$ $8$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[2]$ $[2, 2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t\right) x + 4 t + 2$
2.8.8.11 $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ $2$ $4$ $2$ $8$ $S_4$ (as 8T14) $[4/3, 4/3]$ $[4/3, 4/3]_{3}^{2}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + 2 x + 2$
2.8.8.12 $x^{8} - 2 x^{6} - 2 x^{5} + 8 x^{4} + 8 x^{3} - 4 x + 4$ $2$ $4$ $2$ $8$ $A_4\wr C_2$ (as 8T42) $[4/3, 4/3]$ $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{2} + 2 t x + 2$
2.8.8.13 $x^{8} + 2 x + 2$ $2$ $8$ $1$ $8$ $C_2^3:(C_7: C_3)$ (as 8T36) $[8/7, 8/7, 8/7]$ $[8/7, 8/7, 8/7]_{7}^{3}$ $t + 1$ $x^{8} + 2 x + 2$
2.8.10.1 $x^{8} + 2 x^{3} + 2$ $2$ $8$ $1$ $10$ $C_2^3:(C_7: C_3)$ (as 8T36) $[10/7, 10/7, 10/7]$ $[10/7, 10/7, 10/7]_{7}^{3}$ $t + 1$ $x^{8} + 2 x^{3} + 2$
2.8.10.2 $x^{8} + 2 x^{3} + 2 x^{2} + 2$ $2$ $8$ $1$ $10$ $\textrm{GL(2,3)}$ (as 8T23) $[4/3, 4/3, 3/2]$ $[4/3, 4/3, 3/2]_{3}^{2}$ $t + 1$ $x^{8} + 2 x^{3} + 2 x^{2} + 2$
2.8.10.3 $x^{8} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2$ $2$ $8$ $1$ $10$ $\textrm{GL(2,3)}$ (as 8T23) $[4/3, 4/3, 3/2]$ $[4/3, 4/3, 3/2]_{3}^{2}$ $t + 1$ $x^{8} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2$
2.8.12.1 $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ $2$ $2$ $4$ $12$ $C_4\times C_2$ (as 8T2) $[3]$ $[3]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(4 t^{3} + 4 t^{2}\right) x + 12 t^{3} + 12 t^{2} + 4 t + 2$
2.8.12.2 $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ $2$ $2$ $4$ $12$ $C_4\times C_2$ (as 8T2) $[3]$ $[3]^{4}$ $t^{4} + t + 1$ $x^{2} + 4 x + 6$
2.8.12.3 $x^{8} - 12 x^{7} + 44 x^{6} + 152 x^{5} - 512 x^{4} + 560 x^{3} + 4432 x^{2} + 9376 x + 21744$ $2$ $2$ $4$ $12$ $C_2^2:C_4$ (as 8T10) $[3]$ $[2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + 4 t^{3} x + 4 t^{3} + 12 t + 2$
2.8.12.4 $x^{8} + 8 x^{6} - 16 x^{5} - 376 x^{4} - 128 x^{3} + 8928 x^{2} + 21056 x + 13008$ $2$ $2$ $4$ $12$ $C_2^3: C_4$ (as 8T20) $[3]$ $[2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + 4 t x + 4 t^{2} + 12 t + 2$
2.8.12.5 $x^{8} + 208 x^{5} - 88 x^{4} - 1344 x^{3} + 13440 x^{2} - 16320 x + 17744$ $2$ $2$ $4$ $12$ $C_8$ (as 8T1) $[3]$ $[3]^{4}$ $t^{4} + t + 1$ $x^{2} + 4 t x + 8 t^{3} + 12 t^{2} + 8 t + 6$
2.8.12.6 $x^{8} - 16 x^{6} + 72 x^{4} + 3664$ $2$ $2$ $4$ $12$ $C_8$ (as 8T1) $[3]$ $[3]^{4}$ $t^{4} + t + 1$ $x^{2} + 8 t^{3} + 2$
2.8.12.7 $x^{8} + 16 x^{7} + 160 x^{6} + 800 x^{5} + 2952 x^{4} + 7104 x^{3} + 12992 x^{2} + 14208 x + 10320$ $2$ $2$ $4$ $12$ $C_8:C_2$ (as 8T7) $[3]$ $[2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(4 t^{2} + 4 t + 4\right) x + 8 t^{3} + 8 t^{2} + 2$
2.8.12.8 $x^{8} - 16 x^{6} + 112 x^{5} + 56 x^{4} - 1280 x^{3} + 10176 x^{2} + 39744 x + 40720$ $2$ $2$ $4$ $12$ $(C_8:C_2):C_2$ (as 8T16) $[3]$ $[2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + 4 t x + 8 t^{3} + 4 t^{2} + 12 t + 2$
2.8.12.9 $x^{8} + 16 x^{7} + 68 x^{6} + 192 x^{5} + 1504 x^{4} + 2560 x^{3} + 12016 x^{2} + 13184 x + 15216$ $2$ $2$ $4$ $12$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[3]$ $[2, 2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(4 t + 4\right) x + 12 t^{3} + 12 t^{2} + 4 t + 2$
2.8.12.10 $x^{8} + 68 x^{6} + 272 x^{5} + 928 x^{4} + 5376 x^{3} + 13680 x^{2} + 15552 x + 92016$ $2$ $2$ $4$ $12$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[3]$ $[2, 2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(4 t^{2} + 4 t\right) x + 12 t^{3} + 12 t^{2} + 6$
2.8.12.11 $x^{8} + 16 x^{7} + 92 x^{6} + 352 x^{5} + 1264 x^{4} + 2304 x^{3} + 6544 x^{2} - 256 x + 12080$ $2$ $2$ $4$ $12$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[3]$ $[2, 2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(4 t + 4\right) x + 4 t^{3} + 12 t^{2} + 4 t + 2$
2.8.12.12 $x^{8} + 92 x^{6} + 144 x^{5} + 1232 x^{4} + 5248 x^{3} + 11152 x^{2} + 34752 x + 49072$ $2$ $2$ $4$ $12$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $[3]$ $[2, 2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(4 t^{2} + 4 t\right) x + 4 t^{3} + 12 t^{2} + 6$
2.8.12.13 $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$ $2$ $4$ $2$ $12$ $D_4$ (as 8T4) $[2, 2]$ $[2, 2]^{2}$ $t^{2} + t + 1$ $x^{4} + 2 x^{3} + 2$
2.8.12.14 $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$ $2$ $4$ $2$ $12$ $D_4$ (as 8T4) $[2, 2]$ $[2, 2]^{2}$ $t^{2} + t + 1$ $x^{4} + 2 x^{3} + 6$
2.8.12.15 $x^{8} + 4 x^{7} + 4 x^{6} + 12$ $2$ $4$ $2$ $12$ $C_2^2:C_4$ (as 8T10) $[2, 2]$ $[2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 x^{3} + 4 t + 2$
2.8.12.16 $x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$ $2$ $4$ $2$ $12$ $A_4\times C_2$ (as 8T13) $[2, 2]$ $[2, 2]^{6}$ $t^{2} + t + 1$ $x^{4} + 2 x^{3} + 2 x^{2} + 6$
2.8.12.17 $x^{8} + 4 x^{7} + 6 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 4$ $2$ $4$ $2$ $12$ $C_2^4:C_6$ (as 8T33) $[2, 2]$ $[2, 2, 2, 2]^{6}$ $t^{2} + t + 1$ $x^{4} + 2 x^{3} + \left(2 t + 2\right) x^{2} + 2$
2.8.12.18 $x^{8} - 2 x^{7} + 4 x^{6} + 8 x^{4} + 4 x^{3} + 28$ $2$ $4$ $2$ $12$ $C_2^4:C_6$ (as 8T33) $[2, 2]$ $[2, 2, 2, 2]^{6}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{3} + 4 t + 6$
2.8.12.19 $x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ $2$ $4$ $2$ $12$ $(C_8:C_2):C_2$ (as 8T16) $[2, 2]$ $[2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{3} + 2 t x^{2} + 4 t + 2$
2.8.12.20 $x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 8 x^{4} - 4 x^{3} - 4 x^{2} + 4$ $2$ $4$ $2$ $12$ $C_2^3: C_4$ (as 8T21) $[2, 2]$ $[2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{3} + 2 t x^{2} + 2$
2.8.12.21 $x^{8} + 2 x^{7} + 2 x^{6} + 4 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 4$ $2$ $4$ $2$ $12$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 2]$ $[2, 2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + \left(2 t + 2\right) x^{3} + 2 t x^{2} + 2$
2.8.12.22 $x^{8} + 2 x^{7} + 8 x^{6} + 4 x^{5} + 4 x^{4} + 12 x^{3} + 12$ $2$ $4$ $2$ $12$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 2]$ $[2, 2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + \left(2 t + 2\right) x^{3} + 2 x^{2} + 4 t + 2$
2.8.12.23 $x^{8} - 2 x^{7} + 8 x^{6} - 4 x^{5} + 8 x^{4} - 4 x^{3} + 8 x^{2} + 4$ $2$ $4$ $2$ $12$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 2]$ $[2, 2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{3} + 2 x^{2} + 2$
2.8.12.24 $x^{8} + 2 x^{7} + 2 x^{6} + 4 x^{5} + 16 x^{4} + 12 x^{3} - 12 x^{2} + 36$ $2$ $4$ $2$ $12$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 2]$ $[2, 2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + \left(2 t + 2\right) x^{3} + 2 t x^{2} + 6$
2.8.12.25 $x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ $2$ $8$ $1$ $12$ $S_4\times C_2$ (as 8T24) $[4/3, 4/3, 2]$ $[4/3, 4/3, 2]_{3}^{2}$ $t + 1$ $x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$
2.8.12.26 $x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 6$ $2$ $8$ $1$ $12$ $S_4\times C_2$ (as 8T24) $[4/3, 4/3, 2]$ $[4/3, 4/3, 2]_{3}^{2}$ $t + 1$ $x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 6$
2.8.12.27 $x^{8} + 2 x^{5} + 2 x^{4} + 2$ $2$ $8$ $1$ $12$ $C_2^3:(C_7: C_3)$ (as 8T36) $[12/7, 12/7, 12/7]$ $[12/7, 12/7, 12/7]_{7}^{3}$ $t + 1$ $x^{8} + 2 x^{5} + 2 x^{4} + 2$
2.8.12.28 $x^{8} + 2 x^{5} + 2$ $2$ $8$ $1$ $12$ $C_2^3:(C_7: C_3)$ (as 8T36) $[12/7, 12/7, 12/7]$ $[12/7, 12/7, 12/7]_{7}^{3}$ $t + 1$ $x^{8} + 2 x^{5} + 2$
2.8.12.29 $x^{8} + 2 x^{5} + 2 x^{2} + 6$ $2$ $8$ $1$ $12$ $\textrm{GL(2,3)}$ (as 8T23) $[4/3, 4/3, 2]$ $[4/3, 4/3, 2]_{3}^{2}$ $t + 1$ $x^{8} + 2 x^{5} + 2 x^{2} + 6$
2.8.12.30 $x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ $2$ $8$ $1$ $12$ $\textrm{GL(2,3)}$ (as 8T23) $[4/3, 4/3, 2]$ $[4/3, 4/3, 2]_{3}^{2}$ $t + 1$ $x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$
2.8.14.1 $x^{8} + 2 x^{7} + 6$ $2$ $8$ $1$ $14$ $A_4\times C_2$ (as 8T13) $[2, 2, 2]$ $[2, 2, 2]^{3}$ $t + 1$ $x^{8} + 2 x^{7} + 6$
2.8.14.2 $x^{8} + 2 x^{7} + 2$ $2$ $8$ $1$ $14$ $A_4\times C_2$ (as 8T13) $[2, 2, 2]$ $[2, 2, 2]^{3}$ $t + 1$ $x^{8} + 2 x^{7} + 2$
2.8.14.3 $x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2$ $2$ $8$ $1$ $14$ $C_2^3 : C_4 $ (as 8T19) $[2, 2, 2]$ $[2, 2, 2]^{4}$ $t + 1$ $x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2$
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