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Properties

Label 2.2.4.12a4.2-1.2.6a
Base 2.2.4.12a4.2
Degree \(2\)
e \(2\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{2} + \left(b_{9} \pi^{5} + b_{7} \pi^{4} + a_{5} \pi^{3}\right) x + c_{10} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.12a4.2
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Absolute Artin slopes: $[2,2,3]$
Swan slopes: $[5]$
Means: $\langle\frac{5}{2}\rangle$
Rams: $(5)$
Field count: $60$ (complete)
Ambiguity: $2$
Mass: $48$
Absolute Mass: $24$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $C_2^5:C_4$ (show 2), $C_2^5:C_4$ (show 2), $C_2^5.D_4$ (show 8), $(C_4\times C_8).D_4$ (show 2), $C_2^2.C_2\wr C_4$ (show 6), $D_4^2:C_4$ (show 4), $C_2^4.Q_{16}$ (show 6), $C_2^5.\OD_{16}$ (show 4), $C_2^3.C_2\wr C_4$ (show 6), $C_2^6:C_8$ (show 2), $Q_8^2:C_8$ (show 4), $C_4^3.D_4$ (show 6), $D_4^2:C_8$ (show 8)
Hidden Artin slopes: $[2,2,\frac{9}{4}]^{4}$ (show 2), $[2,2,3]^{4}$ (show 20), $[2,2,3]^{2}$ (show 20), $[2,2,\frac{5}{2}]^{4}$ (show 6), $[2,2,\frac{9}{4}]^{2}$ (show 2), $[2,2,\frac{5}{2}]^{2}$ (show 4), $[2,2,2]^{4}$ (show 2), $[2,2]^{2}$ (show 4)
Indices of inseparability: $[11,6,4,0]$
Associated inertia: $[2,1]$
Jump Set: $[1,2,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 60

Next   displayed columns for results
Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.2.8.36b8.21 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.\OD_{16}$ (as 16T901) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.22 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.\OD_{16}$ (as 16T901) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.23 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.24 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 14$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.25 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.26 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 14$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.27 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_4^3.D_4$ (as 16T936) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.28 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_4^3.D_4$ (as 16T936) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.29 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^2.C_2\wr C_4$ (as 16T681) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.30 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 14$ $C_2^2.C_2\wr C_4$ (as 16T681) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.31 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^4.Q_{16}$ (as 16T702) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.32 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^4.Q_{16}$ (as 16T702) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.33 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.34 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 14$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.35 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_4$ (as 16T688) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.36 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_4$ (as 16T688) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.37 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.38 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 14$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.39 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b8.40 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 14$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + 1$ $[1, 2, 7, 15]$
2.2.8.36b9.21 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.22 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.23 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.\OD_{16}$ (as 16T901) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.24 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.25 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $C_2^5.D_4$ (as 16T633) $256$ $4$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.26 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^5.\OD_{16}$ (as 16T901) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.27 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_4$ (as 16T688) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.28 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.29 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.30 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_4$ (as 16T688) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.31 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.32 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $D_4^2:C_8$ (as 16T941) $512$ $4$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.33 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_4^3.D_4$ (as 16T936) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.34 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $C_4^3.D_4$ (as 16T936) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.35 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^2.C_2\wr C_4$ (as 16T681) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.36 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^2.C_2\wr C_4$ (as 16T681) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.37 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.38 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^4.Q_{16}$ (as 16T702) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.39 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $C_2^4.Q_{16}$ (as 16T702) $256$ $2$ $[2, 2, 2, 2, 3, 3]^{4}$ $[1,1,1,1,2,2]^{4}$ $[2,2,3]^{2}$ $[1,1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b9.40 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, 3, 3]^{8}$ $[1,1,1,1,2,2]^{8}$ $[2,2,3]^{4}$ $[1,1,2]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + t$ $[1, 2, 7, 15]$
2.2.8.36b10.21 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^4.Q_{16}$ (as 16T702) $256$ $2$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{4}$ $[1,1,1,1,\frac{3}{2},2]^{4}$ $[2,2,\frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.22 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 8 x + 6$ $C_2^4.Q_{16}$ (as 16T702) $256$ $2$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{4}$ $[1,1,1,1,\frac{3}{2},2]^{4}$ $[2,2,\frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.23 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.24 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.25 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^6:C_8$ (as 16T924) $512$ $2$ $[2, 2, 2, 2, 2, 3]^{8}$ $[1,1,1,1,1,2]^{8}$ $[2,2,2]^{4}$ $[1,1,1]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.26 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^5:C_4$ (as 16T283) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2]^{2}$ $[1,1]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.27 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 8 x + 6$ $C_2^5:C_4$ (as 16T227) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2]^{2}$ $[1,1]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.28 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^5:C_4$ (as 16T283) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2]^{2}$ $[1,1]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.29 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 8 x + 6$ $C_2^5:C_4$ (as 16T227) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2]^{2}$ $[1,1]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.30 $( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^6:C_8$ (as 16T924) $512$ $2$ $[2, 2, 2, 2, 2, 3]^{8}$ $[1,1,1,1,1,2]^{8}$ $[2,2,2]^{4}$ $[1,1,1]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
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