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Properties

Label 2.2.4.12a3.1-1.2.6a
Base 2.2.4.12a3.1
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: 2.2.4.12a3.1
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: $52$ (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: $C_4\times A_4$ (show 2), $C_2^2\times A_4$ (show 2), $D_4\times A_4$ (show 2), $D_4.A_4$ (show 8), $(C_2^2\times C_4):A_4$ (show 2), $C_2\wr A_4$ (show 2), $C_2^4:(C_2\times A_4)$ (show 10), $C_2^6.(C_4\times A_4)$ (show 24) (incomplete)
Hidden Artin slopes: $[2,2,3]^{3}$ (show 8), not computed (show 24), $[2,2,2]^{3}$ (show 2), $[\ ]^{3}$ (show 4), $[3]^{3}$ (show 8), $[2]^{3}$ (show 2), $[2,2]^{3}$ (show 4) (incomplete)
Indices of inseparability: $[11,6,4,0]$
Associated inertia: $[3,1]$
Jump Set: $[1,2,7,15]$

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
2.2.8.36b6.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^2\times A_4$ (as 16T58) $48$ $4$ $[2, 2, 3]^{6}$ $[1,1,2]^{6}$ $[\ ]^{3}$ $[\ ]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_4\times A_4$ (as 16T57) $48$ $4$ $[2, 2, 3]^{6}$ $[1,1,2]^{6}$ $[\ ]^{3}$ $[\ ]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.3 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $D_4\times A_4$ (as 16T179) $96$ $2$ $[2, 2, 2, 3]^{6}$ $[1,1,1,2]^{6}$ $[2]^{3}$ $[1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $D_4\times A_4$ (as 16T179) $96$ $2$ $[2, 2, 2, 3]^{6}$ $[1,1,1,2]^{6}$ $[2]^{3}$ $[1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.5 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^2\times A_4$ (as 16T58) $48$ $4$ $[2, 2, 3]^{6}$ $[1,1,2]^{6}$ $[\ ]^{3}$ $[\ ]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.6 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_4\times A_4$ (as 16T57) $48$ $4$ $[2, 2, 3]^{6}$ $[1,1,2]^{6}$ $[\ ]^{3}$ $[\ ]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.7 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.9 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.10 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.11 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.12 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.13 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.14 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.15 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2\wr A_4$ (as 16T427) $192$ $4$ $[2, 2, 2, 2, 3]^{6}$ $[1,1,1,1,2]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.16 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $(C_2^2\times C_4):A_4$ (as 16T426) $192$ $4$ $[2, 2, 2, 2, 3]^{6}$ $[1,1,1,1,2]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.17 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 2, 3]^{6}$ $[1,1,1,1,1,2]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.18 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 2, 3]^{6}$ $[1,1,1,1,1,2]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.19 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2\wr A_4$ (as 16T427) $192$ $4$ $[2, 2, 2, 2, 3]^{6}$ $[1,1,1,1,2]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b6.20 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ $(C_2^2\times C_4):A_4$ (as 16T426) $192$ $4$ $[2, 2, 2, 2, 3]^{6}$ $[1,1,1,1,2]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.36b7.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.3 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.5 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.6 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.7 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.8 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.9 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.10 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.11 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.12 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.13 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.14 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.15 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 3, 3]^{6}$ $[1,1,1,1,2,2]^{6}$ $[2,2,3]^{3}$ $[1,1,2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.17 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.18 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.19 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.20 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.21 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.22 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.23 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.24 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $D_4.A_4$ (as 16T180) $96$ $2$ $[2, 2, 3, 3]^{6}$ $[1,1,2,2]^{6}$ $[3]^{3}$ $[2]^{3}$ $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.25 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.26 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.27 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.28 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.29 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
2.2.8.36b7.30 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^6.(C_4\times A_4)$ (as 16T1512) $3072$ $2$ not computed not computed not computed not computed $[11, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + t$ $[1, 2, 7, 15]$
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