Properties

Label 2.8.18.61
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(18\)
Galois group $C_2^3 : C_4 $ (as 8T19)

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Discriminant root field: $\Q_{2}$
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:\( x^{8} + 6 x^{6} + 4 x^{3} + 2 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2^2.D_4$ (as 8T19)
Inertia group:$C_2^3$
Wild inertia group:$C_2^3$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:[2, 2, 3]
Galois mean slope:$9/4$
Galois splitting model:$x^{8} - 4 x^{6} + 30 x^{4} + 20 x^{2} + 25$