Properties

Label 2.8.18.16
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(18\)
Galois group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)

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Defining polynomial

\(x^{8} + 6 x^{6} + 4 x^{2} + 4\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $18$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2,7/2]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 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:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + \left(10 t + 6\right) x^{2} + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display
Indices of inseparability:$[6, 2, 0]$

Invariants of the Galois closure

Galois group:$C_2\wr C_2^2$ (as 8T29)
Inertia group:Intransitive group isomorphic to $C_2^2\wr C_2$
Wild inertia group:$C_2^2\wr C_2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2, 2, 3, 7/2, 7/2]$
Galois mean slope:$51/16$
Galois splitting model:$x^{8} + 2 x^{6} - 26 x^{4} - 72 x^{2} + 36$