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Properties

Label 2.1.2.3a1.3-1.4.16b
Base 2.1.2.3a1.3
Degree \(4\)
e \(4\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{4} + b_{15} \pi^{4} x^{3} + \left(c_{18} \pi^{5} + b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{17} \pi^{5} + a_{13} \pi^{4}\right) x + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[3,4,\frac{17}{4}]$
Swan slopes: $[4,\frac{9}{2}]$
Means: $\langle2,\frac{13}{4}\rangle$
Rams: $(4,5)$
Field count: $16$ (complete)
Ambiguity: $4$
Mass: $16$
Absolute Mass: $8$

Diagrams

Varying

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

Galois group: $C_2^3: C_4$ (show 8), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 8)
Hidden Artin slopes: $[2,\frac{7}{2}]$ (show 8), $[2,\frac{7}{2}]^{2}$ (show 8)
Indices of inseparability: $[21,16,8,0]$
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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Showing all 16

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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
2.1.8.28b1.33 $x^{8} + 8 x^{5} + 4 x^{4} + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.34 $x^{8} + 8 x^{5} + 4 x^{4} + 18$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.35 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.36 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 18$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.37 $x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.38 $x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 16 x^{2} + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.39 $x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 18$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.40 $x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 16 x^{2} + 18$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.41 $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.42 $x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.43 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.44 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 2$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.45 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.46 $x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 24 x^{2} + 18$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.47 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.28b1.48 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 18$ $C_2^3: C_4$ (as 8T20) $32$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$ $[1,2,\frac{5}{2},3,\frac{13}{4}]$ $[2,\frac{7}{2}]$ $[1,\frac{5}{2}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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