Properties

Label 2.6.6.2
Base \(\Q_{2}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(6\)
Galois group $A_4\times C_2$ (as 6T6)

Related objects

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Defining polynomial

\( x^{6} - x^{4} - 5 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $6$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $3$
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

2.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + 2 x + 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times A_4$ (as 6T6)
Inertia group:Intransitive group isomorphic to $C_2^2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$3/2$
Galois splitting model:$x^{6} - 3 x^{4} + 3$