Properties

Label 2.8.21.8
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(21\)
Galois group $C_2 \wr C_2\wr C_2$ (as 8T35)

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Defining polynomial

\(x^{8} + 24 x^{2} + 8\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $21$
Discriminant root field: $\Q_{2}(\sqrt{2})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{-1})$, 2.4.6.8

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:\( x^{8} + 18 x^{6} + 108 x^{4} + 216 x^{2} + 2 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$D_4^2.C_2$ (as 8T35)
Inertia group:$(((C_4 \times C_2): C_2):C_2):C_2$
Wild inertia group:$C_2\wr C_2^2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:[2, 2, 3, 7/2, 7/2, 15/4]
Galois mean slope:$111/32$
Galois splitting model:$x^{8} - 2 x^{6} - 4 x^{4} + 20 x^{2} + 50$