Properties

Label 2.8.22.3
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(22\)
Galois group $Q_8$ (as 8T5)

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Defining polynomial

\(x^{8} + 8 x^{7} + 60 x^{6} + 136 x^{5} + 256 x^{4} + 240 x^{3} + 104 x^{2} + 112 x + 76\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $22$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 2 }) }$: $8$
This field is Galois over $\Q_{2}.$
Visible slopes:$[3, 4]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.4.6.2

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: \( x^{4} + \left(8 t + 8\right) x^{3} + \left(12 t + 4\right) x^{2} + \left(8 t + 8\right) x + 4 t + 10 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Not computed

Invariants of the Galois closure

Galois group: $Q_8$ (as 8T5)
Inertia group: Intransitive group isomorphic to $C_4$
Wild inertia group: $C_4$
Unramified degree: $2$
Tame degree: $1$
Wild slopes: $[3, 4]$
Galois mean slope: $11/4$
Galois splitting model:$x^{8} - 60 x^{6} + 810 x^{4} - 1800 x^{2} + 900$