Properties

Label 2.2.6.20a
Base 2.1.1.0a1.1
Degree \(12\)
e \(6\)
f \(2\)
c \(20\)

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Defining polynomial over unramified subextension

$x^{6} + 2 a_{5} x^{5} + 4 c_{10} x^{4} + 4 b_{9} x^{3} + 4 b_{7} x + 2 d_{0}$

Invariants

Residue field characteristic: $2$
Degree: $12$
Base field: $\Q_{2}$
Ramification index $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $20$
Artin slopes: $[\frac{8}{3}]$
Swan slopes: $[\frac{5}{3}]$
Means: $\langle\frac{5}{6}\rangle$
Rams: $(5)$
Field count: $52$ (complete)
Ambiguity: $12$
Mass: $48$
Absolute Mass: $24$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[1,1]$
Jump Set: $[3,9]$

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing 1-50 of 52

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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
2.2.6.20a1.1 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{12}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{12}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{6}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.2 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4\wr C_2$ (as 12T126) $288$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.3 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4\wr C_2$ (as 12T126) $288$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.4 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{12}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{12}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{6}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.5 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.6 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.7 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.8 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.9 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.10 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.11 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.12 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 2\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.13 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4\wr C_2$ (as 12T126) $288$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.14 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{12}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{12}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{6}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.15 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{12}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{12}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{6}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.16 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 ) + 2 x$ $A_4\wr C_2$ (as 12T126) $288$ $2$ $[\frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.17 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.18 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.19 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.20 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.21 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.22 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.23 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.24 16 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x$ $A_4^2:D_4$ (as 12T208) $1152$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.25 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_2^2$ (as 12T158) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.26 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.27 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_2^2$ (as 12T158) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.28 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.29 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_2^2$ (as 12T158) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.30 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.31 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_2^2$ (as 12T158) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.32 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + \left(4 x + 6\right) ( x^{2} + x + 1 ) + 2 x$ $A_4^2:C_4$ (as 12T159) $576$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{3}$ $[1,\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{3}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.33 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $\GL(2,\mathbb{Z}/4)$ (as 12T50) $96$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,2,\frac{8}{3}]$ $[1,1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.34 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $\GL(2,\mathbb{Z}/4)$ (as 12T50) $96$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,2,\frac{8}{3}]$ $[1,1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.35 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2\wr S_3$ (as 12T135) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]$ $[\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.36 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2\wr S_3$ (as 12T135) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]$ $[\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.37 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 ) + 2$ $\GL(2,\mathbb{Z}/4)$ (as 12T50) $96$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,2,\frac{8}{3}]$ $[1,1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.38 4 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 ) + 2$ $\GL(2,\mathbb{Z}/4)$ (as 12T50) $96$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,2,\frac{8}{3}]$ $[1,1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.39 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2$ $C_2\times S_4$ (as 12T21) $48$ $4$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,\frac{8}{3}]$ $[1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.40 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 2$ $A_4:C_4$ (as 12T30) $48$ $4$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,\frac{8}{3}]$ $[1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.41 1 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^3:S_4$ (as 12T106) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3}]$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.42 1 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^3.S_4$ (as 12T107) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{8}{3}]$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.43 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 2$ $C_2\times S_4$ (as 12T21) $48$ $4$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,\frac{8}{3}]$ $[1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.44 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ $A_4:C_4$ (as 12T30) $48$ $4$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[2,\frac{8}{3}]$ $[1,\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.45 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 ) + 2$ $S_4$ (as 12T9) $24$ $4$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{8}{3}]$ $[\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.46 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 2$ $A_4:C_4$ (as 12T30) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{4}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{4}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.47 1 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^2:S_4$ (as 12T69) $96$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{8}{3}]$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.48 1 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^3.S_4$ (as 12T107) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{4}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]_{3}^{4}$ $[\frac{4}{3},\frac{4}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.49 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $S_4$ (as 12T9) $24$ $4$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{8}{3}]$ $[\frac{5}{3}]$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
2.2.6.20a1.50 2 $( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$ $A_4:C_4$ (as 12T30) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{4}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{4}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[5, 0]$ $[1, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 9]$
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