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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 (16 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
2.2.8.56a1.43 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 x^{6} + 8 x^{5} + 8 t x^{4} + 16 t x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.44 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 24 x ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 x^{6} + 8 x^{5} + 24 t x^{4} + 16 t x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.45 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 x^{6} + 8 x^{5} + 8 t x^{4} + \left(16 t + 16\right) x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.46 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 24 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 x^{6} + 8 x^{5} + 24 t x^{4} + \left(16 t + 16\right) x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.59 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + 4 x^{6} + 8 x^{5} + 8 t x^{4} + \left(16 t + 16\right) x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.60 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 24 x ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + 4 x^{6} + 8 x^{5} + 24 t x^{4} + \left(16 t + 16\right) x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.61 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + 4 x^{6} + 8 x^{5} + 8 t x^{4} + 16 t x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.62 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 24 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + 4 x^{6} + 8 x^{5} + 24 t x^{4} + 16 t x^{3} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.75 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 16 t x^{3} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.76 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 16 t x^{4} + 16 t x^{3} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.77 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.78 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 16 t x^{4} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.79 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 24 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + \left(16 t + 16\right) x^{3} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.80 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 24 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 16 t x^{4} + \left(16 t + 16\right) x^{3} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.81 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 16 x^{3} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.82 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $56$ $\OD_{16}.D_4$ (as 16T215) $2$ $1$ $[3, \frac{7}{2}, \frac{9}{2}]$ $[2,\frac{5}{2},\frac{7}{2}]$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 8 x^{5} + 16 t x^{4} + 16 x^{3} + 8 t x^{2} + 2$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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