Properties

Label 2.8.10.2
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(10\)
Galois group $\textrm{GL(2,3)}$ (as 8T23)

Related objects

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Defining polynomial

\( x^{8} + 20 x^{2} + 20 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

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:\( x^{8} + 26 x^{7} + 412 x^{6} + 4188 x^{5} + 25300 x^{4} + 88302 x^{3} + 166662 x^{2} + 135904 x + 54766 \)

Invariants of the Galois closure

Galois group:$\GL(2,3)$ (as 8T23)
Inertia group:$\SL(2,3)$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 3/2]
Galois mean slope:$4/3$
Galois splitting model:$x^{8} - 6 x^{4} + 4 x^{2} - 3$