Properties

Label 2.8.14.14
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(14\)
Galois group $C_2 \wr S_4$ (as 8T44)

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Defining polynomial

\(x^{8} + 2 x^{7} + 4 x^{3} + 2 x^{2} + 4 x + 6\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[4/3, 4/3, 5/2]$

Intermediate fields

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:$\Q_{2}$
Relative Eisenstein polynomial: \( x^{8} + 2 x^{7} + 4 x^{3} + 2 x^{2} + 4 x + 6 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{2} + 1$
Associated inertia:$1$,$1$
Indices of inseparability:$[7, 2, 2, 0]$

Invariants of the Galois closure

Galois group:$C_2\wr S_4$ (as 8T44)
Inertia group:$C_2\wr A_4$ (as 8T38)
Wild inertia group:$C_2\wr C_2^2$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:$[4/3, 4/3, 2, 7/3, 7/3, 5/2]$
Galois mean slope:$223/96$
Galois splitting model:$x^{8} + 4 x^{2} + 12$