Properties

Label 2.4.8.6
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)
Galois group $D_{4}$ (as 4T3)

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Defining polynomial

\(x^{4} + 6 x^{2} + 4 x + 10\) 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: $\Q_{2}(\sqrt{5})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2, 3]$

Intermediate fields

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

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: \( x^{4} + 6 x^{2} + 4 x + 10 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$D_4$ (as 4T3)
Inertia group:$C_2^2$ (as 4T2)
Wild inertia group:$C_2^2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2, 3]$
Galois mean slope:$2$
Galois splitting model: $x^{4} + 6 x^{2} + 4 x + 2$ Copy content Toggle raw display