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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
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) $[4/3, 4/3]$ $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{2} + 2 t x + 2$
2.8.12.13 $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$ $2$ $4$ $2$ $12$ $D_4$ (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) $[2, 2]$ $[2, 2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + \left(2 t + 2\right) x^{3} + 2 t x^{2} + 6$
2.8.16.1 $x^{8} + 8 x^{7} + 24 x^{6} + 40 x^{5} + 60 x^{4} + 32 x^{3} + 88 x^{2} + 108$ $2$ $4$ $2$ $16$ $C_4\times C_2$ (as 8T2) $[2, 3]$ $[2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(4 t + 6\right) x^{2} + 4 x + 12 t + 6$
2.8.16.10 $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 12 x^{4} + 8 x^{3} + 52 x^{2} + 72 x + 108$ $2$ $4$ $2$ $16$ $D_4\times C_2$ (as 8T9) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 2\right) x^{2} + 4 t x + 12 t + 6$
2.8.16.11 $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ $2$ $4$ $2$ $16$ $D_4\times C_2$ (as 8T9) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(2 t + 2\right) x^{2} + 4 t x + 4 t + 10$
2.8.16.12 $x^{8} + 8 x^{7} + 14 x^{6} - 12 x^{5} + 12 x^{4} + 40 x^{3} + 124 x^{2} + 72 x + 108$ $2$ $4$ $2$ $16$ $C_2^2:C_4$ (as 8T10) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(6 t + 2\right) x^{2} + 4 t x + 12 t + 6$
2.8.16.13 $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 8 x^{4} + 24 x^{3} + 20 x^{2} - 8 x + 4$ $2$ $4$ $2$ $16$ $D_4\times C_2$ (as 8T9) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 2\right) x^{2} + 4 t x + 2$
2.8.16.14 $x^{8} + 10 x^{6} - 4 x^{5} + 40 x^{4} - 8 x^{3} + 100 x^{2} + 24 x + 84$ $2$ $4$ $2$ $16$ $C_2^2:C_4$ (as 8T10) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(2 t + 6\right) x^{2} + 4 t x + 8 t + 10$
2.8.16.15 $x^{8} + 8 x^{7} + 18 x^{6} + 12 x^{5} + 52 x^{4} + 72 x^{3} + 60 x^{2} + 40 x + 28$ $2$ $4$ $2$ $16$ $C_2^2:C_4$ (as 8T10) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(6 t + 4\right) x^{2} + \left(4 t + 4\right) x + 4 t + 6$
2.8.16.16 $x^{8} + 8 x^{7} + 26 x^{6} + 36 x^{5} + 40 x^{4} + 104 x^{3} + 156 x^{2} - 56 x + 196$ $2$ $4$ $2$ $16$ $D_4\times C_2$ (as 8T9) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 6\right) x^{2} + 4 t x + 14$
2.8.16.17 $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} + 12 x^{4} + 104 x^{3} + 52 x^{2} - 24 x + 156$ $2$ $4$ $2$ $16$ $C_2^2:C_4$ (as 8T10) $[2, 3]$ $[2, 2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 2\right) x^{2} + 4 t x + 4 t + 14$
2.8.16.18 $x^{8} + 12 x^{6} + 4 x^{5} + 52 x^{4} + 24 x^{3} + 112 x^{2} + 56 x + 76$ $2$ $4$ $2$ $16$ $C_2^3: C_4$ (as 8T20) $[2, 3]$ $[2, 3, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 6 x^{2} + \left(4 t + 4\right) x + 4 t + 10$
2.8.16.19 $x^{8} + 8 x^{7} + 16 x^{6} - 4 x^{5} + 12 x^{4} + 88 x^{3} + 40 x^{2} - 8 x + 76$ $2$ $4$ $2$ $16$ $C_2^3: C_4$ (as 8T20) $[2, 3]$ $[2, 3, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(4 t + 2\right) x^{2} + 4 t x + 4 t + 10$
2.8.16.2 $x^{8} + 4 x^{6} + 8 x^{5} + 16 x^{4} + 16 x^{3} + 40 x^{2} + 48 x + 84$ $2$ $4$ $2$ $16$ $C_4\times C_2$ (as 8T2) $[2, 3]$ $[2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + 4 x + 8 t + 10$
2.8.16.20 $x^{8} - 4 x^{6} + 8 x^{5} + 20 x^{4} - 16 x^{3} + 8 x^{2} + 16 x + 4$ $2$ $4$ $2$ $16$ $V_4^2:S_3$ (as 8T34) $[8/3, 8/3]$ $[4/3, 4/3, 8/3, 8/3]_{3}^{2}$ $t^{2} + t + 1$ $x^{4} + 4 t x^{2} + 4 x + 2$
2.8.16.21 $x^{8} + 12 x^{6} - 4 x^{5} + 56 x^{4} - 24 x^{3} + 136 x^{2} + 8 x + 148$ $2$ $4$ $2$ $16$ $Q_8:C_2$ (as 8T11) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 6 x^{2} + 4 t x + 8 t + 14$
2.8.16.22 $x^{8} + 8 x^{7} + 20 x^{6} + 12 x^{5} + 40 x^{3} + 40 x^{2} - 24 x + 36$ $2$ $4$ $2$ $16$ $Q_8:C_2$ (as 8T11) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 2 x^{2} + 4 t x + 6$
2.8.16.23 $x^{8} + 4 x^{6} + 4 x^{5} + 16 x^{4} + 8 x^{3} + 40 x^{2} + 72 x + 84$ $2$ $4$ $2$ $16$ $Q_8:C_2$ (as 8T11) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + \left(4 t + 4\right) x + 8 t + 10$
2.8.16.24 $x^{8} + 8 x^{7} + 28 x^{6} + 44 x^{5} + 24 x^{4} - 8 x^{3} + 40 x^{2} - 8 x + 4$ $2$ $4$ $2$ $16$ $Q_8:C_2$ (as 8T11) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 6 x^{2} + 4 t x + 2$
2.8.16.25 $x^{8} - 4 x^{5} + 8 x^{4} + 16 x^{2} + 8 x + 28$ $2$ $4$ $2$ $16$ $A_4\wr C_2$ (as 8T42) $[8/3, 8/3]$ $[8/3, 8/3, 8/3, 8/3]_{3}^{6}$ $t^{2} + t + 1$ $x^{4} + 4 t x + 4 t + 6$
2.8.16.26 $x^{8} + 8 x^{7} + 24 x^{6} + 36 x^{5} + 44 x^{4} + 64 x^{3} + 64 x^{2} + 24 x + 36$ $2$ $4$ $2$ $16$ $A_4\wr C_2$ (as 8T42) $[8/3, 8/3]$ $[8/3, 8/3, 8/3, 8/3]_{3}^{6}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 4 x^{2} + \left(4 t + 4\right) x + 6$
2.8.16.27 $x^{8} + 8 x^{7} + 12 x^{6} - 20 x^{5} + 12 x^{4} + 80 x^{3} - 8 x^{2} - 24 x + 36$ $2$ $4$ $2$ $16$ $A_4\wr C_2$ (as 8T42) $[8/3, 8/3]$ $[8/3, 8/3, 8/3, 8/3]_{3}^{6}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 4 t x^{2} + 4 t x + 6$
2.8.16.28 $x^{8} - 4 x^{5} + 16 x^{2} + 24 x + 12$ $2$ $4$ $2$ $16$ $A_4\wr C_2$ (as 8T42) $[8/3, 8/3]$ $[8/3, 8/3, 8/3, 8/3]_{3}^{6}$ $t^{2} + t + 1$ $x^{4} + 4 t x + 4 t + 2$
2.8.16.29 $x^{8} + 4 x^{6} + 8 x^{5} + 4 x^{4} + 16 x^{3} + 16 x^{2} + 12$ $2$ $4$ $2$ $16$ $(C_8:C_2):C_2$ (as 8T16) $[2, 3]$ $[2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + 4 x + 4 t + 2$
2.8.16.3 $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$ $2$ $4$ $2$ $16$ $C_4\times C_2$ (as 8T2) $[2, 3]$ $[2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(4 t + 2\right) x^{2} + 4 x + 12 t + 2$
2.8.16.30 $x^{8} - 4 x^{7} + 28 x^{6} - 16 x^{5} + 24 x^{4} + 88 x^{3} + 40 x^{2} + 16 x + 52$ $2$ $4$ $2$ $16$ $C_2^3: C_4$ (as 8T21) $[2, 3]$ $[2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 4 t x^{3} + 6 x^{2} + 4 x + 8 t + 6$
2.8.16.31 $x^{8} + 8 x^{7} + 26 x^{6} + 36 x^{5} + 8 x^{4} - 24 x^{3} + 20 x^{2} + 56 x + 52$ $2$ $4$ $2$ $16$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 3]$ $[2, 2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 6\right) x^{2} + 4 t x + 8 t + 2$
2.8.16.32 $x^{8} + 8 x^{7} + 26 x^{6} + 36 x^{5} + 20 x^{4} + 24 x^{3} + 68 x^{2} + 8 x + 28$ $2$ $4$ $2$ $16$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 3]$ $[2, 2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 6\right) x^{2} + 4 t x + 4 t + 6$
2.8.16.33 $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} + 8 x^{4} + 88 x^{3} + 60 x^{2} + 8 x + 148$ $2$ $4$ $2$ $16$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 3]$ $[2, 2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 2\right) x^{2} + 4 t x + 8 t + 14$
2.8.16.34 $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 20 x^{4} - 24 x^{3} + 44 x^{2} + 88 x + 124$ $2$ $4$ $2$ $16$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $[2, 3]$ $[2, 2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(2 t + 2\right) x^{2} + 4 t x + 12 t + 2$
2.8.16.35 $x^{8} + 8 x^{6} - 4 x^{5} + 44 x^{4} + 8 x^{3} + 152 x^{2} + 40 x + 172$ $2$ $4$ $2$ $16$ $C_8:C_2$ (as 8T7) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 6\right) x^{2} + 4 t x + 12 t + 14$
2.8.16.36 $x^{8} - 4 x^{5} + 4 x^{4} + 24 x^{3} + 88 x^{2} + 88 x + 124$ $2$ $4$ $2$ $16$ $C_8:C_2$ (as 8T7) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 2\right) x^{2} + 4 t x + 12 t + 2$
2.8.16.37 $x^{8} + 8 x^{6} - 4 x^{5} + 28 x^{4} + 8 x^{3} + 88 x^{2} + 72 x + 108$ $2$ $4$ $2$ $16$ $C_8:C_2$ (as 8T7) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 6\right) x^{2} + 4 t x + 12 t + 6$
2.8.16.38 $x^{8} - 4 x^{5} + 20 x^{4} + 24 x^{3} + 88 x^{2} + 56 x + 124$ $2$ $4$ $2$ $16$ $C_8:C_2$ (as 8T7) $[2, 3]$ $[2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 2\right) x^{2} + 4 t x + 12 t + 10$
2.8.16.39 $x^{8} + 4 x^{6} - 4 x^{5} + 32 x^{4} - 8 x^{3} + 72 x^{2} - 56 x + 196$ $2$ $4$ $2$ $16$ $C_2^3: C_4$ (as 8T21) $[2, 3]$ $[2, 3, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + 4 t x + 14$
2.8.16.4 $x^{8} + 12 x^{7} + 58 x^{6} + 160 x^{5} + 329 x^{4} + 500 x^{3} + 408 x^{2} + 68 x + 61$ $2$ $4$ $2$ $16$ $D_4$ (as 8T4) $[2, 3]$ $[2, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 10$
2.8.16.40 $x^{8} + 8 x^{6} - 4 x^{5} + 56 x^{4} + 8 x^{3} + 128 x^{2} - 56 x + 196$ $2$ $4$ $2$ $16$ $C_2^3: C_4$ (as 8T21) $[2, 3]$ $[2, 3, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 6\right) x^{2} + 4 t x + 14$
2.8.16.41 $x^{8} - 2 x^{6} + 8 x^{5} + 32 x^{4} - 8 x^{3} + 12 x^{2} + 16 x + 4$ $2$ $4$ $2$ $16$ $Z_8 : Z_8^\times$ (as 8T15) $[2, 3]$ $[2, 2, 3, 3]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(6 t + 2\right) x^{2} + 4 x + 2$
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