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Properties

Label 2.1.4.10a1.2-2.2.18a
Base 2.1.4.10a1.2
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: 2.1.4.10a1.2
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $18$
Absolute Artin slopes: $[3,\frac{7}{2},\frac{19}{4}]$
Swan slopes: $[8]$
Means: $\langle4\rangle$
Rams: $(8)$
Field count: $144$ (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: $(C_2^2\times D_4):C_4$ (show 4), $C_2^4.D_4$ (show 8), $C_2^4.D_4$ (show 4), $(C_2^3\times C_4):C_4$ (show 8), $(C_2^2\times D_4):C_4$ (show 4), $C_2^5:C_4$ (show 8), $C_2^4.D_4$ (show 4), $(C_2\times C_4^2):C_4$ (show 8), $C_2^6:(C_2\times C_4)$ (show 16), $C_2^4.(C_4\times D_4)$ (show 16), $C_2^5:\SD_{16}$ (show 16), $C_2^5:D_8$ (show 16), $C_2^5.(C_2\times D_4)$ (show 32)
Hidden Artin slopes: $[2,4,\frac{17}{4}]$ (show 48), $[2,2,4,\frac{17}{4},\frac{9}{2}]$ (show 64), $[2,2,\frac{7}{2},4,\frac{17}{4}]$ (show 32)
Indices of inseparability: $[22,14,8,0]$
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 144

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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
2.2.8.58a1.1 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 2$ $C_2^5:C_4$ (as 16T283) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.2 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 2$ $(C_2^3\times C_4):C_4$ (as 16T280) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.3 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 2$ $C_2^4.D_4$ (as 16T287) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.4 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 2$ $(C_2^2\times D_4):C_4$ (as 16T281) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.5 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 2$ $C_2^5:C_4$ (as 16T283) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.6 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 2$ $(C_2^3\times C_4):C_4$ (as 16T280) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.7 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.8 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.9 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.10 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.11 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.12 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.13 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.14 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,2,\frac{7}{2},4,\frac{17}{4}]$ $[1,1,\frac{5}{2},3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.15 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:C_4$ (as 16T283) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.16 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 2$ $(C_2^3\times C_4):C_4$ (as 16T280) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.17 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^4.D_4$ (as 16T287) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.18 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $(C_2^2\times D_4):C_4$ (as 16T281) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.19 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:C_4$ (as 16T283) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.20 $( x^{2} + x + 1 )^{8} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $(C_2^3\times C_4):C_4$ (as 16T280) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,4,\frac{17}{4}]$ $[1,3,\frac{13}{4}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.21 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.22 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.23 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.24 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.25 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.26 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.27 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.28 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.29 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.30 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.31 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.32 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.33 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.34 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^5.(C_2\times D_4)$ (as 16T1000) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.35 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.36 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.37 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.38 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.39 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.40 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 2$ $C_2^5:D_8$ (as 16T988) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.41 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.42 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.43 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.44 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.45 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.46 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.47 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.48 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{5} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.49 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.58a1.50 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 x ( x^{2} + x + 1 ) + 2$ $C_2^5:\SD_{16}$ (as 16T970) $512$ $2$ $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}]^{2}$ $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4}]^{2}$ $[2,2,4,\frac{17}{4},\frac{9}{2}]$ $[1,1,3,\frac{13}{4},\frac{7}{2}]$ $[22, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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