Properties

Label 2.4.8.3
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 6 x^{2} + 4 x + 14\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
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:$[2, 3]$

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial: \( x^{4} + 6 x^{2} + 4 x + 14 \) Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_2^2$ (as 4T2)
Inertia group: $C_2^2$ (as 4T2)
Wild inertia group: $C_2^2$
Unramified degree: $1$
Tame degree: $1$
Wild slopes: $[2, 3]$
Galois mean slope: $2$
Galois splitting model:$x^{4} - 4 x^{2} + 1$