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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
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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
2.2.8.54a1.27 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 x^{7} + 2 x^{4} + 16 t x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.30 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 x^{7} + \left(16 t + 2\right) x^{4} + 16 t x^{3} + 16 t x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.74 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + 8 x^{5} + \left(16 t + 2\right) x^{4} + 16 t x^{3} + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.574 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + 4 t x^{6} + 8 t x^{5} + \left(8 t + 2\right) x^{4} + 16 t x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.616 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + \left(4 t + 4\right) x^{6} + 8 t x^{5} + \left(24 t + 2\right) x^{4} + 16 t x^{3} + 16 x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.618 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + \left(4 t + 4\right) x^{6} + 8 t x^{5} + \left(8 t + 2\right) x^{4} + 16 x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.1116 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + 2 x^{4} + 16 t x^{3} + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.1121 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + 4 x^{6} + \left(16 t + 2\right) x^{4} + 16 t x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.1215 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + \left(16 t + 2\right) x^{4} + 16 t x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.1217 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + \left(16 t + 2\right) x^{4} + 16 t x^{3} + 16 t x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.1580 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 16 t x^{3} + 8 x^{2} + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.1747 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 2\right) ( x^{2} + x + 1 )^{4} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + 4 t x^{6} + 8 t x^{5} + \left(24 t + 2\right) x^{4} + 16 t x^{3} + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.2126 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + \left(8 t + 8\right) x^{7} + \left(4 t + 4\right) x^{6} + \left(8 t + 8\right) x^{5} + 2 x^{4} + 16 x^{3} + 8 x^{2} + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.2130 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + \left(24 x + 2\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 16 x ( x^{2} + x + 1 ) + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{2}$ $[2,3,\frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 t x^{6} + 8 t x^{5} + \left(16 t + 2\right) x^{4} + 16 x^{3} + 8 x^{2} + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.2165 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 t x^{6} + \left(8 t + 8\right) x^{5} + \left(16 t + 2\right) x^{4} + 16 t x^{3} + 8 x^{2} + 16 x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
2.2.8.54a1.2168 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(24 x + 2\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $2$ $2$ $8$ $54$ $C_4^2:D_4$ (as 16T340) $2$ $1$ $[2, \frac{7}{2}, \frac{9}{2}]$ $[1,\frac{5}{2},\frac{7}{2}]$ $[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $t^{2} + t + 1$ $x^{8} + 8 t x^{7} + 4 t x^{6} + \left(8 t + 8\right) x^{5} + \left(16 t + 2\right) x^{4} + 8 x^{2} + 16 x + 2$ $[20, 12, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 5, 13, 21]$
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