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Properties

Label 2.2.8.40c
Base 2.1.1.0a1.1
Degree \(16\)
e \(8\)
f \(2\)
c \(40\)

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Defining polynomial over unramified subextension

$x^{8} + 4 b_{15} x^{7} + 4 a_{13} x^{5} + 4 b_{12} x^{4} + 4 a_{10} x^{2} + 8 c_{16} + 2$

Invariants

Residue field characteristic: $2$
Degree: $16$
Base field: $\Q_{2}$
Ramification index $e$: $8$
Residue field degree $f$: $2$
Discriminant exponent $c$: $40$
Artin slopes: $[\frac{8}{3},\frac{8}{3},3]$
Swan slopes: $[\frac{5}{3},\frac{5}{3},2]$
Means: $\langle\frac{5}{6},\frac{5}{4},\frac{13}{8}\rangle$
Rams: $(\frac{5}{3},\frac{5}{3},3)$
Field count: $148$ (complete)
Ambiguity: $4$
Mass: $144$
Absolute Mass: $72$

Diagrams

Varying

Indices of inseparability: $[13,10,8,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7,15]$

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 148

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
2.2.8.40c1.1 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2\times S_4$ (as 16T61) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.2 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $A_4:C_4$ (as 16T62) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.3 4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $\GL(2,\mathbb{Z}/4)$ (as 16T193) $96$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2]_{3}$ $[1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.4 4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $\GL(2,\mathbb{Z}/4)$ (as 16T193) $96$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2]_{3}$ $[1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.5 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2\times S_4$ (as 16T61) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.6 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $A_4:C_4$ (as 16T62) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.7 4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2\wr S_3$ (as 16T765) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2]_{3}$ $[\frac{1}{3},\frac{1}{3},1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.8 4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $C_2\wr S_3$ (as 16T765) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2]_{3}$ $[\frac{1}{3},\frac{1}{3},1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.9 2 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^3:S_4$ (as 16T429) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.10 2 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $C_2^3.S_4$ (as 16T434) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.11 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2\wr S_3$ (as 16T765) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2]_{3}$ $[\frac{1}{3},\frac{1}{3},1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.12 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $C_2\wr S_3$ (as 16T765) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2]_{3}$ $[\frac{1}{3},\frac{1}{3},1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.13 2 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^3:S_4$ (as 16T429) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.14 2 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $C_2^3.S_4$ (as 16T434) $192$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.15 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2\times S_4$ (as 16T61) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.16 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $A_4:C_4$ (as 16T62) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.17 4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $\GL(2,\mathbb{Z}/4)$ (as 16T193) $96$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2]_{3}$ $[1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.18 4 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $\GL(2,\mathbb{Z}/4)$ (as 16T193) $96$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[2]_{3}$ $[1]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.19 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2\times S_4$ (as 16T61) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.20 4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $A_4:C_4$ (as 16T62) $48$ $4$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\ ]_{3}$ $[\ ]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c2.1 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.2 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.3 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.4 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.5 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.6 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.7 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.8 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.9 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.10 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.11 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.12 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.13 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.14 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.15 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.16 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.17 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.18 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.19 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.20 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.21 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.22 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.23 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.24 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4.S_4$ (as 16T750) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.25 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.26 16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.27 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.28 16 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.29 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.40c2.30 16 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 10$ $C_2^4:S_4$ (as 16T751) $384$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},3]_{3}$ $[\frac{1}{3},\frac{1}{3},2]_{3}$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7, 15]$
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