Properties

Label 2.8.8.2
Base \(\Q_{2}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(8\)
Galois group $C_2^2:C_4$ (as 8T10)

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Defining polynomial

\( x^{8} + 2 x^{7} + 8 x^{2} + 48 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$|\Aut(K/\Q_{ 2 })|$: $4$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.4.0.1, 2.4.4.3, 2.4.4.4

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

Unramified/totally ramified tower

Unramified subfield:2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + \left(6 t^{3} + 4 t^{2} + 2 t\right) x + 4 t^{3} + 2 t^{2} + 6 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2^2:C_4$ (as 8T10)
Inertia group:Intransitive group isomorphic to $C_2^2$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$3/2$
Galois splitting model:$x^{8} - 2 x^{6} + 4 x^{4} - 3 x^{2} + 1$