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Properties

Label 2.1.4.8b1.2-1.4.20c
Base 2.1.4.8b1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{4} + \left(b_{27} \pi^{7} + b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + \left(b_{10} \pi^{3} + a_{6} \pi^{2}\right) x^{2} + \left(b_{25} \pi^{7} + b_{21} \pi^{6} + a_{17} \pi^{5}\right) x + c_{28} \pi^{8} + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.8b1.2
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Absolute Artin slopes: $[2,3,3,4]$
Swan slopes: $[3,7]$
Means: $\langle\frac{3}{2},\frac{17}{4}\rangle$
Rams: $(3,11)$
Field count: $44$ (complete)
Ambiguity: $4$
Mass: $64$
Absolute Mass: $16$

Diagrams

Varying

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

Galois group: $C_4^2:C_2$ (show 2), $C_4 \times D_4$ (show 2), $C_4^2:C_2$ (show 3), $C_4^2:C_2$ (show 1), $C_2^2.D_4$ (show 4), $C_8.D_4$ (show 2), $C_2^4.C_2^3$ (show 4), $C_4^2.D_4$ (show 4), $C_4^2.D_4$ (show 4), $\OD_{16}.D_4$ (show 2), $C_2^5.(C_2\times D_4)$ (show 8), $(D_4\times C_2^3).C_2^3$ (show 8)
Hidden Artin slopes: $[\ ]^{2}$ (show 12), $[\frac{7}{2}]^{2}$ (show 2), $[2,2,3]^{4}$ (show 8), $[2,2,\frac{7}{2}]^{4}$ (show 8), $[2]^{4}$ (show 4), $[2,\frac{7}{2}]^{2}$ (show 6), $[\frac{7}{2}]^{4}$ (show 4)
Indices of inseparability: $[37,26,24,8,0]$
Associated inertia: $[1,2,1]$
Jump Set: $[1,5,10,32,48]$

Fields

Galois group Galois Artin Indices Hidden content
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
2.1.16.52k1.38 $x^{16} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 10$ $C_4^2.D_4$ (as 16T321) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,2,2,\frac{5}{2},3]^{4}$ $[\frac{7}{2}]^{4}$ $[\frac{5}{2}]^{4}$ $[37, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.52k1.39 $x^{16} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 26$ $C_4^2.D_4$ (as 16T321) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,2,2,\frac{5}{2},3]^{4}$ $[\frac{7}{2}]^{4}$ $[\frac{5}{2}]^{4}$ $[37, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.52k1.40 $x^{16} + 8 x^{15} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 10$ $C_4^2.D_4$ (as 16T321) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,2,2,\frac{5}{2},3]^{4}$ $[\frac{7}{2}]^{4}$ $[\frac{5}{2}]^{4}$ $[37, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.52k1.44 $x^{16} + 8 x^{15} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 10$ $C_4^2.D_4$ (as 16T321) $128$ $4$ $[2, 3, 3, \frac{7}{2}, 4]^{4}$ $[1,2,2,\frac{5}{2},3]^{4}$ $[\frac{7}{2}]^{4}$ $[\frac{5}{2}]^{4}$ $[37, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
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