Properties

Label 7.8.6.1
Base \(\Q_{7}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $Q_8$ (as 8T5)

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Defining polynomial

\(x^{8} + 14 x^{4} - 245\) 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: $\Q_{7}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 7 }) }$: $8$
This field is Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 7.4.2.1

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: \( x^{4} + 7 t + 28 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{3} + 4z^{2} + 6z + 4$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$Q_8$ (as 8T5)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{8} - 3 x^{7} + 22 x^{6} - 60 x^{5} + 201 x^{4} - 450 x^{3} + 1528 x^{2} - 3069 x + 4561$