Properties

Label 2.1.6.6a
Base 2.1.1.0a1.1
Degree \(6\)
e \(6\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{6} + 2 c_{2} x^{2} + 2 a_{1} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}$
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Artin slopes: $[\frac{4}{3}]$
Swan slopes: $[\frac{1}{3}]$
Means: $\langle\frac{1}{6}\rangle$
Rams: $(1)$
Field count: $2$ (complete)
Ambiguity: $2$
Mass: $1$
Absolute Mass: $1$

Diagrams

Varying

Indices of inseparability: $[1,0]$
Associated inertia: $[2,1]$
Jump Set: $[3,7]$

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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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
2.1.6.6a1.1 1 $x^{6} + 2 x + 2$ $S_4$ (as 6T8) $24$ $2$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $[1, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 7]$
2.1.6.6a1.2 1 $x^{6} + 2 x^{2} + 2 x + 2$ $S_4$ (as 6T7) $24$ $2$ $[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3}]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $[1, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + 1$ $[3, 7]$
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