Base \(\Q_{2}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(4\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13\) Copy content Toggle raw display


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

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$

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

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$1$
Galois splitting model: $x^{4} + 8 x^{2} + 4$ Copy content Toggle raw display