Properties

Label 2.8.18.11
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

\(x^{8} + 8 x^{7} + 10 x^{6} - 16 x^{5} + 80 x^{4} + 96 x^{3} + 100 x^{2} + 144 x + 84\) 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

$\Q_{2}(\sqrt{5})$, 2.4.4.3

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: \( x^{4} + 4 x^{3} + 6 t x^{2} + \left(8 t + 8\right) x + 8 t + 10 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

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:$x^{8} - 2 x^{6} - 4 x^{2} + 4$