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Degree $n$ Residue characteristic $p$ Discriminant exponent $c$ Galois group
Ramification index $e$ Residue field degree $f$ Galois tame degree $t$ Galois unram. degree $u$
Num. automorphisms Inertia subgroup Indices Assoc. Inertia Jump Set
Top Artin slope Galois Artin Visible Artin Wild inertia subgroup
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Results (24 matches)

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Label Polynomial $p$ $f$ $e$ $c$ Galois group $u$ $t$ Visible Artin slopes Visible Swan slopes Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Unram. Ext. Eisen. Poly. Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.4.2.8a5.2 $( x^{4} + x + 1 )^{2} + \left(2 x^{3} + 2 x + 2\right) ( x^{4} + x + 1 ) + 4 x^{2} + 2$ $2$ $4$ $2$ $8$ $C_2^3: C_4$ (as 8T20) $4$ $1$ $[2]$ $[1]$ $[2, 2, 2]^{4}$ $[1,1,1]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{2} + \left(2 t^{3} + 2 t^{2} + 2\right) x + 2$ $[1, 0]$ $[1]$ $z + (t^3 + t + 1)$ $[1, 3]$
2.4.2.12a1.5 $( x^{4} + x + 1 )^{2} + 4 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $2$ $12$ $C_2^3: C_4$ (as 8T20) $4$ $1$ $[3]$ $[2]$ $[2, 2, 3]^{4}$ $[1,1,2]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{2} + \left(4 t^{2} + 4\right) x + 2$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.4.16b2.9 $( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 4 x + 2$ $2$ $2$ $4$ $16$ $C_2^3: C_4$ (as 8T20) $4$ $1$ $[2, 3]$ $[1,2]$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[3]^{2}$ $[2]^{2}$ $t^{2} + t + 1$ $x^{4} + \left(4 t + 4\right) x^{3} + 2 x^{2} + \left(4 t + 4\right) x + 4 t + 2$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 2, 8]$
2.2.4.16b2.10 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 4 x + 2$ $2$ $2$ $4$ $16$ $C_2^3: C_4$ (as 8T20) $4$ $1$ $[2, 3]$ $[1,2]$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[3]^{2}$ $[2]^{2}$ $t^{2} + t + 1$ $x^{4} + 2 x^{2} + 4 t x + 4 t + 2$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 2, 8]$
2.2.4.20a1.5 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $2$ $2$ $4$ $20$ $C_2^3: C_4$ (as 8T20) $2$ $1$ $[3, \frac{7}{2}]$ $[2,\frac{5}{2}]$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 8 t + 2$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a1.6 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 x + 2$ $2$ $2$ $4$ $20$ $C_2^3: C_4$ (as 8T20) $2$ $1$ $[3, \frac{7}{2}]$ $[2,\frac{5}{2}]$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 8 t x + 8 t + 2$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a1.17 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $2$ $2$ $4$ $20$ $C_2^3: C_4$ (as 8T20) $2$ $1$ $[3, \frac{7}{2}]$ $[2,\frac{5}{2}]$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + 4 x^{2} + 8 t + 2$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a1.18 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x + 2$ $2$ $2$ $4$ $20$ $C_2^3: C_4$ (as 8T20) $2$ $1$ $[3, \frac{7}{2}]$ $[2,\frac{5}{2}]$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $t^{2} + t + 1$ $x^{4} + 4 x^{3} + \left(8 t + 4\right) x^{2} + 8 t + 2$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.8.28b1.41 $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.42 $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.43 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.44 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 2$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 2$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.45 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.46 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 18$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 18$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.47 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.48 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.57 $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 10$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 10$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.58 $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 26$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 26$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.59 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 10$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 10$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.60 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 10$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 10$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.61 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 26$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 26$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.62 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 26$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 26$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.63 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 10$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 10$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.64 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 26$ $2$ $1$ $8$ $28$ $C_2^3: C_4$ (as 8T20) $1$ $1$ $[3, 4, \frac{17}{4}]$ $[2,3,\frac{13}{4}]$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $t + 1$ $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 26$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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