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Properties

Label 2.1.4.11a1.5-2.2.18a
Base 2.1.4.11a1.5
Degree \(4\)
e \(2\)
f \(2\)
c \(18\)

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Defining polynomial over unramified subextension

$x^{2} + \left(b_{15} \pi^{8} + b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5}\right) x + c_{16} \pi^{9} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.5
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $18$
Absolute Artin slopes: $[3,4,5]$
Swan slopes: $[8]$
Means: $\langle4\rangle$
Rams: $(8)$
Field count: $152$ (complete)
Ambiguity: $4$
Mass: $256$
Absolute Mass: $64$

Diagrams

Varying

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

Galois group: $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $C_2\times \SD_{16}$ (show 2), $C_4^2:C_2^2$ (show 4), $C_4^2:C_2^2$ (show 8), $\OD_{16}:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 8), $C_4^2.D_4$ (show 8), $C_4^2:D_4$ (show 4), $C_2\wr D_4$ (show 8), $C_2^5.(C_2\times D_4)$ (show 16), $C_2^4.(C_4\times D_4)$ (show 16), $C_2^7.(C_2\times D_4)$ (show 64) (incomplete)
Hidden Artin slopes: $[2]$ (show 8), $[2,2,\frac{7}{2}]$ (show 4), $[2,\frac{7}{2},\frac{17}{4}]$ (show 24), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ (show 8), $[2,3,\frac{7}{2},4,\frac{17}{4}]$ (show 16), not computed (show 56), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ (show 16), $[2,\frac{7}{2}]$ (show 12), $[2,3,\frac{7}{2}]$ (show 8) (incomplete)
Indices of inseparability: $[24,16,8,0]$
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 152

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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
2.2.8.62a1.197 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x + 2$ $C_4^2:D_4$ (as 16T342) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.198 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x + 2$ $C_2\wr D_4$ (as 16T401) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.199 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x + 2$ $C_2\wr D_4$ (as 16T401) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.200 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x + 2$ $C_4^2:D_4$ (as 16T345) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.201 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x + 2$ $C_4^2:D_4$ (as 16T359) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.202 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x + 2$ $C_4^2:D_4$ (as 16T345) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.203 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.204 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.205 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.206 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.207 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.208 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.209 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.210 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.211 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.212 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.213 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.214 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2^5.(C_2\times D_4)$ (as 16T813) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, 5]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},4]^{2}$ $[2,3,\frac{7}{2},4,\frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.215 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2\wr D_4$ (as 16T401) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.216 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_4^2:D_4$ (as 16T342) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.217 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_2\wr D_4$ (as 16T401) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.218 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_4^2:D_4$ (as 16T359) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.219 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_4^2:D_4$ (as 16T345) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.220 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 16 x + 2$ $C_4^2:D_4$ (as 16T345) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.221 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.222 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.223 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.224 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.225 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.226 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.227 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.228 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.229 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.230 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.231 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.232 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.233 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.234 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.235 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{13}{4},\frac{15}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.236 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.237 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.238 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.239 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.240 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.241 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.242 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.243 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.244 $( x^{2} + x + 1 )^{8} + \left(16 x + 24\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.245 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.246 $( x^{2} + x + 1 )^{8} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^7.(C_2\times D_4)$ (as 16T1481) $2048$ $2$ not computed not computed not computed not computed $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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