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Degree Residue characteristic Galois group Galois unramified degree Assoc. Inertia
Ramification index Discriminant exponent Inertia subgroup Galois tame degree Indices
Residue field degree Top slope Wild Visible Wild inertia subgroup
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Results (24 matches)

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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
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) $[2]$ $[2, 2, 2]^{4}$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t + 2\right) x + 4 t^{2} + 4 t + 2$
2.8.12.19 $x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ $2$ $4$ $2$ $12$ $(C_8:C_2):C_2$ (as 8T16) $[2, 2]$ $[2, 2, 2]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 t x^{3} + 2 t x^{2} + 4 t + 2$
2.8.12.8 $x^{8} - 16 x^{6} + 112 x^{5} + 56 x^{4} - 1280 x^{3} + 10176 x^{2} + 39744 x + 40720$ $2$ $2$ $4$ $12$ $(C_8:C_2):C_2$ (as 8T16) $[3]$ $[2, 2, 3]^{4}$ $t^{4} + t + 1$ $x^{2} + 4 t x + 8 t^{3} + 4 t^{2} + 12 t + 2$
2.8.16.29 $x^{8} + 4 x^{6} + 8 x^{5} + 4 x^{4} + 16 x^{3} + 16 x^{2} + 12$ $2$ $4$ $2$ $16$ $(C_8:C_2):C_2$ (as 8T16) $[2, 3]$ $[2, 2, 3]^{4}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + 4 x + 4 t + 2$
2.8.22.39 $x^{8} + 8 x^{6} + 16 x^{5} + 44 x^{4} + 64 x^{3} + 176 x^{2} + 224 x + 628$ $2$ $4$ $2$ $22$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4]$ $[2, 2, 3, 4]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{2} + 8 x + 24 t + 26$
2.8.22.40 $x^{8} + 4 x^{6} - 8 x^{5} + 24 x^{4} + 32 x^{3} + 152 x^{2} + 112 x + 124$ $2$ $4$ $2$ $22$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4]$ $[2, 2, 3, 4]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 4\right) x^{2} + 8 t x + 12 t + 10$
2.8.22.41 $x^{8} + 4 x^{6} - 8 x^{5} + 40 x^{4} + 32 x^{3} + 280 x^{2} + 240 x + 732$ $2$ $4$ $2$ $22$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4]$ $[2, 2, 3, 4]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 4\right) x^{2} + 8 t x + 28 t + 26$
2.8.22.42 $x^{8} + 8 x^{6} + 16 x^{5} + 28 x^{4} + 64 x^{3} + 112 x^{2} + 96 x + 84$ $2$ $4$ $2$ $22$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4]$ $[2, 2, 3, 4]^{2}$ $t^{2} + t + 1$ $x^{4} + 4 x^{2} + 8 x + 8 t + 10$
2.8.31.113 $x^{8} + 4 x^{4} + 42$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 42$
2.8.31.114 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 58$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 58$
2.8.31.115 $x^{8} + 4 x^{4} + 58$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 58$
2.8.31.116 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 10$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 10$
2.8.31.117 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 42$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 42$
2.8.31.118 $x^{8} + 4 x^{4} + 10$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 10$
2.8.31.119 $x^{8} + 4 x^{4} + 26$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 26$
2.8.31.120 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 26$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 26$
2.8.31.29 $x^{8} + 4 x^{4} + 50$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 50$
2.8.31.30 $x^{8} + 4 x^{4} + 18$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 18$
2.8.31.31 $x^{8} + 4 x^{4} + 34$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 34$
2.8.31.32 $x^{8} + 4 x^{4} + 2$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 4 x^{4} + 2$
2.8.31.33 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 2$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 2$
2.8.31.34 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 50$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 50$
2.8.31.35 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 18$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 18$
2.8.31.36 $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 34$ $2$ $8$ $1$ $31$ $(C_8:C_2):C_2$ (as 8T16) $[3, 4, 5]$ $[2, 3, 7/2, 4, 5]$ $t + 1$ $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 34$
  displayed columns for results