Properties

Label 7.8.6.3
Base \(\Q_{7}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

\(x^{8} - 154 x^{4} - 1421\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 7 }) }$: $4$
This field is not Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{3})$, 7.4.2.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 35 t + 28 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $\OD_{16}$ (as 8T7)
Inertia group: Intransitive group isomorphic to $C_4$
Wild inertia group: $C_1$
Unramified degree: $4$
Tame degree: $4$
Wild slopes: None
Galois mean slope: $3/4$
Galois splitting model:$x^{8} - x^{7} - 13 x^{6} - 13 x^{5} + 25 x^{4} + 38 x^{3} - 33 x^{2} - 34 x + 11$