Properties

Label 2.6.6.8
Base \(\Q_{2}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(6\)
Galois group $S_4$ (as 6T8)

Related objects

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Defining polynomial

\( x^{6} + 2 x + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $6$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.3.2.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{6} + 2 x + 2 \)

Invariants of the Galois closure

Galois group:$S_4$ (as 6T8)
Inertia group:$A_4$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3]
Galois mean slope:$7/6$
Galois splitting model:$x^{6} - 12 x^{3} + 18 x + 30$