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Properties

Label 2.2.4.20a2.17-1.2.8a
Base 2.2.4.20a2.17
Degree \(2\)
e \(2\)
f \(1\)
c \(8\)

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Defining polynomial

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

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.20a2.17
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[3,\frac{7}{2},\frac{9}{2}]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $192$ (complete)
Ambiguity: $2$
Mass: $192$
Absolute Mass: $96$

Diagrams

Varying

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

Galois group: $C_8^2:C_2^2$ (show 4), $D_8:\SD_{16}$ (show 4), $C_8^2:(C_2\times C_4)$ (show 8), $C_4^3.D_4$ (show 16), $\OD_{16}^2:(C_2\times C_4)$ (show 32), $C_2^7.D_4:D_4$ (show 64), $C_2^7.C_2\wr C_4$ (show 64) (incomplete)
Hidden Artin slopes: not computed (show 144), $[2,2,3,\frac{7}{2},\frac{9}{2}]$ (show 16), $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ (show 16), $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ (show 8), $[2,2,\frac{7}{2},\frac{9}{2}]$ (show 8) (incomplete)
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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Showing 1-50 of 192

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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
2.2.8.56a3.1025 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 2$ $D_8:\SD_{16}$ (as 16T706) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1026 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 2$ $D_8:\SD_{16}$ (as 16T706) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1027 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $D_8:\SD_{16}$ (as 16T706) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1028 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $D_8:\SD_{16}$ (as 16T706) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1029 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1030 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1031 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1032 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1033 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 2$ $C_8^2:C_2^2$ (as 16T687) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1034 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 2$ $C_8^2:C_2^2$ (as 16T687) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1035 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_8^2:C_2^2$ (as 16T687) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1036 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_8^2:C_2^2$ (as 16T687) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,\frac{7}{2},\frac{9}{2}]$ $[1,1,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1037 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1038 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1039 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1040 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1041 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1042 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1043 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1044 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $C_8^2:(C_2\times C_4)$ (as 16T892) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{4}$ $[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{4}$ $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{7}{2}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1045 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1046 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1047 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + \left(16 x + 16\right) ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1048 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + \left(16 x + 16\right) ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1049 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1050 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1051 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1052 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1053 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1054 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 ) + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1055 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1056 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1057 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1058 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1059 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1060 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1061 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1062 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1063 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1064 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 20 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1129 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1130 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 28 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1131 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 24 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1132 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 28 ( x^{2} + x + 1 )^{4} + 24 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1133 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1134 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 28 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1135 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 24 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1136 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 28 ( x^{2} + x + 1 )^{4} + 24 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $\OD_{16}^2:(C_2\times C_4)$ (as 16T1434) $2048$ $2$ not computed not computed not computed not computed $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1137 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 12 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.56a3.1138 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 28 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 2$ $C_4^3.D_4$ (as 16T1011) $512$ $2$ $[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]^{2}$ $[2,2,3,\frac{7}{2},\frac{9}{2}]$ $[1,1,2,\frac{5}{2},\frac{7}{2}]$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 3, 7, 15]$
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