Properties

Label 2.6.4.2
Base \(\Q_{2}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $S_3\times C_3$ (as 6T5)

Related objects

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Defining polynomial

\(x^{6} - 2 x^{3} + 4\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $3$
This field is not Galois over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{5})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} - 2 t \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 6T5)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} - 2 x^{3} + 4$