Properties

Label 2.8.12.30
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(12\)
Galois group $\textrm{GL(2,3)}$ (as 8T23)

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Defining polynomial

\(x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
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:$[4/3, 4/3, 2]$

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: \( x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$\GL(2,3)$ (as 8T23)
Inertia group:$\SL(2,3)$ (as 8T12)
Wild inertia group:$Q_8$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:$[4/3, 4/3, 2]$
Galois mean slope:$19/12$
Galois splitting model:$x^{8} - 6 x^{4} - 4 x^{2} - 3$