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Properties

Label 2.2.8.32c
Base 2.1.1.0a1.1
Degree \(16\)
e \(8\)
f \(2\)
c \(32\)

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Defining polynomial over unramified subextension

$x^{8} + 2 a_{6} x^{6} + \left(2 b_{4} + 4 c_{12}\right) x^{4} + 4 b_{11} x^{3} + 4 a_{9} x + 4 c_{8} + 2$

Invariants

Residue field characteristic: $2$
Degree: $16$
Base field: $\Q_{2}$
Ramification index $e$: $8$
Residue field degree $f$: $2$
Discriminant exponent $c$: $32$
Artin slopes: $[2,2,\frac{5}{2}]$
Swan slopes: $[1,1,\frac{3}{2}]$
Means: $\langle\frac{1}{2},\frac{3}{4},\frac{9}{8}\rangle$
Rams: $(1,1,3)$
Field count: $167$ (complete)
Ambiguity: $16$
Mass: $144$
Absolute Mass: $72$

Diagrams

Varying

Indices of inseparability: $[9,6,4,0]$ (show 122), $[9,6,6,0]$ (show 45)
Associated inertia: $[1,1]$ (show 21), $[2,1]$ (show 84), $[3,1]$ (show 62)
Jump Set: $[1,2,7,15]$ (show 42), $[1,3,6,16]$ (show 23), $[1,3,7,15]$ (show 76), $[1,3,9,17]$ (show 26)

Galois groups and Hidden Artin slopes

Select desired size of Galois group. Note that the following data has not all been computed for fields in this family, so the tables below are incomplete.

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 167

Next   displayed columns for results
Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.2.8.32c1.1 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 ) + 2$ $C_4\wr C_2$ (as 16T42) $32$ $8$ $[2, 2, \frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.2 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $C_4^2:C_4$ (as 16T174) $64$ $4$ $[2, 2, 2, \frac{5}{2}]^{4}$ $[1,1,1,\frac{3}{2}]^{4}$ $[2]^{2}$ $[1]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.3 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $QD_{16}$ (as 16T12) $16$ $16$ $[2, 2, \frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[\ ]$ $[\ ]$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.4 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_{16}$ (as 16T14) $16$ $16$ $[2, 2, \frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[\ ]$ $[\ ]$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.5 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x + 2$ $C_4^2.C_4$ (as 16T139) $64$ $8$ $[2, 2, 2, \frac{5}{2}]^{4}$ $[1,1,1,\frac{3}{2}]^{4}$ $[2]^{2}$ $[1]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 6, 16]$
2.2.8.32c1.6 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + 4 x + 2$ $C_4^2.C_4$ (as 16T139) $64$ $8$ $[2, 2, 2, \frac{5}{2}]^{4}$ $[1,1,1,\frac{3}{2}]^{4}$ $[2]^{2}$ $[1]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 6, 16]$
2.2.8.32c1.7 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + 4 x + 2$ $C_4\wr C_2$ (as 16T28) $32$ $4$ $[2, 2, \frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 6, 16]$
2.2.8.32c1.8 1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 4 x + 2$ $C_4^2:C_4$ (as 16T154) $64$ $4$ $[2, 2, 2, \frac{5}{2}]^{4}$ $[1,1,1,\frac{3}{2}]^{4}$ $[2]^{2}$ $[1]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 6, 16]$
2.2.8.32c1.9 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x + 2$ $C_4\wr C_2$ (as 16T28) $32$ $4$ $[2, 2, \frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 6, 16]$
2.2.8.32c1.10 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 ) + 6$ $C_4\wr C_2$ (as 16T42) $32$ $8$ $[2, 2, \frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.11 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 6$ $C_4^2:C_4$ (as 16T174) $64$ $4$ $[2, 2, 2, \frac{5}{2}]^{4}$ $[1,1,1,\frac{3}{2}]^{4}$ $[2]^{2}$ $[1]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.12 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 6$ $QD_{16}$ (as 16T12) $16$ $16$ $[2, 2, \frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[\ ]$ $[\ ]$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c1.13 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 6$ $Q_{16}$ (as 16T14) $16$ $16$ $[2, 2, \frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[\ ]$ $[\ ]$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 9, 17]$
2.2.8.32c2.1 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^4.Q_{16}$ (as 16T684) $256$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{4}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{4}$ $[2,2,\frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 9, 17]$
2.2.8.32c2.2 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^4.Q_{16}$ (as 16T684) $256$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{4}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{4}$ $[2,2,\frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 9, 17]$
2.2.8.32c2.3 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^3.C_2\wr C_4$ (as 16T828) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{8}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 6, 16]$
2.2.8.32c2.4 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^3.C_2\wr C_4$ (as 16T828) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{8}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 6, 16]$
2.2.8.32c2.5 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 ) + 6$ $C_2^4.Q_{16}$ (as 16T684) $256$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{4}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{4}$ $[2,2,\frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 9, 17]$
2.2.8.32c2.6 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 ) + 6$ $C_2^4.Q_{16}$ (as 16T684) $256$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{4}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{4}$ $[2,2,\frac{5}{2}]^{2}$ $[1,1,\frac{3}{2}]^{2}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 9, 17]$
2.2.8.32c2.7 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 4 x + 6$ $C_2^3.C_2\wr C_4$ (as 16T828) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{8}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 6, 16]$
2.2.8.32c2.8 4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 4 x + 6$ $C_2^3.C_2\wr C_4$ (as 16T828) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{8}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[9, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 6, 16]$
2.2.8.32c3.1 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.2 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.3 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.4 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.5 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.6 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.7 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c3.8 8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $Q_8^2:C_6$ (as 16T730) $384$ $2$ $[2, 2, 2, 2, \frac{5}{2}, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2},\frac{3}{2}]^{6}$ $[2,2,\frac{5}{2}]^{3}$ $[1,1,\frac{3}{2}]^{3}$ $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32c4.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.3 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.5 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.6 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.7 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c4.8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32c5.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.3 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.4 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.5 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.6 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.7 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c5.8 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^2.C_2^8:C_{12}$ (as 16T1754) $12288$ $2$ not computed not computed not computed not computed $[9, 6, 6, 0]$ $[3, 1]$ $z^6 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32c6.1 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 2$ $C_2\wr A_4$ (as 16T427) $192$ $4$ $[2, 2, 2, 2, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2}]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[9, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.32c6.2 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 2$ $(C_2^2\times C_4):A_4$ (as 16T426) $192$ $4$ $[2, 2, 2, 2, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2}]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[9, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.32c6.3 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 2, \frac{5}{2}]^{6}$ $[1,1,1,1,1,\frac{3}{2}]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[9, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.32c6.4 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^4:(C_2\times A_4)$ (as 16T716) $384$ $2$ $[2, 2, 2, 2, 2, \frac{5}{2}]^{6}$ $[1,1,1,1,1,\frac{3}{2}]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[9, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
2.2.8.32c6.5 2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $C_2\wr A_4$ (as 16T427) $192$ $4$ $[2, 2, 2, 2, \frac{5}{2}]^{6}$ $[1,1,1,1,\frac{3}{2}]^{6}$ $[2,2]^{3}$ $[1,1]^{3}$ $[9, 6, 4, 0]$ $[3, 1]$ $z^6 + z^2 + 1,z + 1$ $[1, 2, 7, 15]$
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