Properties

Label 2.8.16.37
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(16\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

\(x^{8} + 8 x^{6} - 4 x^{5} + 28 x^{4} + 8 x^{3} + 88 x^{2} + 72 x + 108\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $16$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $4$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2, 3]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.4.4.2

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: \( x^{4} + \left(4 t + 6\right) x^{2} + 4 t x + 12 t + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t$,$z^{2} + 1$
Associated inertia:$1$,$1$
Indices of inseparability:$[5, 2, 0]$

Invariants of the Galois closure

Galois group:$\OD_{16}$ (as 8T7)
Inertia group:Intransitive group isomorphic to $C_2\times C_4$
Wild inertia group:$C_2\times C_4$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2, 3, 3]$
Galois mean slope:$5/2$
Galois splitting model:$x^{8} + 10 x^{6} + 25 x^{4} + 20 x^{2} + 5$