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Properties

Label 2.1.4.10a1.2-2.2.16a
Base 2.1.4.10a1.2
Degree \(4\)
e \(2\)
f \(2\)
c \(16\)

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Defining polynomial over unramified subextension

$x^{2} + \left(b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5} + a_{7} \pi^{4}\right) x + c_{14} \pi^{8} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.10a1.2
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[3,\frac{7}{2},\frac{9}{2}]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $112$ (complete)
Ambiguity: $4$
Mass: $192$
Absolute Mass: $48$

Diagrams

Varying

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

Galois group: $C_2\times D_8$ (show 1), $Q_{16}:C_2$ (show 1), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $D_8:C_2$ (show 1), $C_2\times \SD_{16}$ (show 1), $D_4:D_4$ (show 4), $D_4.D_4$ (show 4), $\OD_{16}:D_4$ (show 8), $(C_2\times C_4^2):C_4$ (show 8), $C_2^4.(C_4\times D_4)$ (show 16), $C_2^5:\SD_{16}$ (show 16), $C_2^5:D_8$ (show 16), $C_2^5.(C_2\times D_4)$ (show 32)
Hidden Artin slopes: $[2]$ (show 8), $[2,3,4]$ (show 16), $[2,2,\frac{7}{2},\frac{15}{4},\frac{9}{2}]$ (show 64), $[2,2]$ (show 8), $[2,2,2,\frac{7}{2}]^{2}$ (show 8), $[2,2,3,4]^{2}$ (show 8)
Indices of inseparability: $[21,14,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
2.2.8.56a1.1 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2$ $D_4:D_4$ (as 16T142) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.2 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 2$ $D_4.D_4$ (as 16T162) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.3 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $D_4:D_4$ (as 16T142) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $D_4.D_4$ (as 16T162) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.5 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.6 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.7 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.8 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.9 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.10 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.11 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.12 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.13 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.14 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.15 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.16 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.17 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.18 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.19 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.20 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.21 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2$ $D_4:D_4$ (as 16T142) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.22 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 2$ $D_4.D_4$ (as 16T162) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.23 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $D_4:D_4$ (as 16T142) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.24 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $D_4.D_4$ (as 16T162) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2,2]$ $[1,1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.25 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.26 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.27 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.28 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.29 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 2$ $C_2\times D_8$ (as 16T29) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.30 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $D_8:C_2$ (as 16T44) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.31 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $D_8:C_2$ (as 16T45) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.32 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $D_8:C_2$ (as 16T38) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.33 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.34 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $(C_2\times C_4^2):C_4$ (as 16T291) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.35 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.36 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}:D_4$ (as 16T251) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,\frac{7}{2}]^{2}$ $[2,3,4]$ $[1,2,3]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.149 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.150 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.151 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.152 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.153 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.154 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.155 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{2} + 2$ $D_8:C_2$ (as 16T38) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.156 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 2$ $Q_{16}:C_2$ (as 16T32) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.157 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $D_8:C_2$ (as 16T44) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.158 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2\times \SD_{16}$ (as 16T48) $32$ $4$ $[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$ $[1,2,\frac{5}{2},\frac{7}{2}]^{2}$ $[2]$ $[1]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.159 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a1.160 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.(C_4\times D_4)$ (as 16T869) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, 4, \frac{9}{2}]^{4}$ $[1,1,2,2,\frac{5}{2},3,\frac{7}{2}]^{4}$ $[2,2,3,4]^{2}$ $[1,1,2,3]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.56a2.1 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{15}{4},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{11}{4},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56a2.2 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{4} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{15}{4},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{11}{4},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
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