Properties

Label 2.1.2.3a1.1-1.4.19a
Base 2.1.2.3a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{4} + \left(b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + \left(b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + b_{17} \pi^{5}\right) x + c_{24} \pi^{7} + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{-2})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Absolute Artin slopes: $[3,4,5]$
Swan slopes: $[4,6]$
Means: $\langle2,4\rangle$
Rams: $(4,8)$
Field count: $72$ (complete)
Ambiguity: $4$
Mass: $64$
Absolute Mass: $32$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $D_{8}$ (show 6), $QD_{16}$ (show 2), $Z_8 : Z_8^\times$ (show 8), $C_4\wr C_2$ (show 16), $(C_4^2 : C_2):C_2$ (show 8), $C_2 \wr C_2\wr C_2$ (show 32)
Hidden Artin slopes: $[2]$ (show 4), $[2]^{2}$ (show 8), $[2,\frac{7}{2}]$ (show 16), $[2,\frac{7}{2},\frac{17}{4}]^{2}$ (show 32), $[\ ]^{2}$ (show 4), $[2,\frac{7}{2}]^{2}$ (show 8)
Indices of inseparability: $[24,16,8,0]$
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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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
2.1.8.31a1.1 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 34$ $D_{8}$ (as 8T6) $16$ $2$ $[2, 3, 4, 5]$ $[1,2,3,4]$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.2 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 2$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]^{2}$ $[1]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.3 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 34$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]^{2}$ $[1]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.4 $x^{8} + 16 x^{7} + 8 x^{6} + 34$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.5 $x^{8} + 8 x^{6} + 34$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.6 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x + 34$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.7 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x + 34$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.8 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.9 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.10 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 34$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.11 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 34$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.12 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 50$ $D_{8}$ (as 8T6) $16$ $2$ $[2, 3, 4, 5]$ $[1,2,3,4]$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.13 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 18$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]^{2}$ $[1]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.14 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 50$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]^{2}$ $[1]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.15 $x^{8} + 16 x^{7} + 8 x^{6} + 50$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.16 $x^{8} + 8 x^{6} + 50$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.17 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.18 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.19 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.20 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.21 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.22 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 16 x + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.23 $x^{8} + 8 x^{6} + 2$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.24 $x^{8} + 16 x^{7} + 8 x^{6} + 2$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.25 $x^{8} + 8 x^{6} + 16 x^{5} + 2$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.26 $x^{8} + 8 x^{6} + 16 x^{5} + 34$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.27 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 2$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.28 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 34$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.29 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 2$ $D_{8}$ (as 8T6) $16$ $2$ $[2, 3, 4, 5]$ $[1,2,3,4]$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.30 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.31 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x + 2$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.32 $x^{8} + 8 x^{6} + 18$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.33 $x^{8} + 16 x^{7} + 8 x^{6} + 18$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.34 $x^{8} + 8 x^{6} + 16 x^{5} + 18$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.35 $x^{8} + 8 x^{6} + 16 x^{5} + 50$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.36 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 18$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.37 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 50$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, \frac{7}{2}, 4, 5]$ $[1,2,\frac{5}{2},3,4]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.38 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 18$ $D_{8}$ (as 8T6) $16$ $2$ $[2, 3, 4, 5]$ $[1,2,3,4]$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.39 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.40 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.41 $x^{8} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.42 $x^{8} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.43 $x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.44 $x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.45 $x^{8} + 16 x^{7} + 16 x^{5} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.46 $x^{8} + 16 x^{5} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.47 $x^{8} + 16 x^{5} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.48 $x^{8} + 16 x^{7} + 16 x^{5} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.49 $x^{8} + 16 x^{7} + 8 x^{2} + 16 x + 50$ $D_{8}$ (as 8T6) $16$ $2$ $[3, 4, 5]^{2}$ $[2,3,4]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.50 $x^{8} + 8 x^{2} + 16 x + 50$ $D_{8}$ (as 8T6) $16$ $2$ $[3, 4, 5]^{2}$ $[2,3,4]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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