Properties

Label 2.8.8.11
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(8\)
Galois group $S_4$ (as 8T14)

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Defining polynomial

\(x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9\) Copy content Toggle raw display

Invariants

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

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.4.4.5

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

Ramification polygon

Residual polynomials:$z + 1$
Associated inertia:$1$
Indices of inseparability:$[1, 1, 0]$

Invariants of the Galois closure

Galois group:$S_4$ (as 8T14)
Inertia group:Intransitive group isomorphic to $A_4$
Wild inertia group:$C_2^2$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:$[4/3, 4/3]$
Galois mean slope:$7/6$
Galois splitting model:$x^{8} - 4 x^{7} + 4 x^{6} - 2 x^{5} + 5 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$