Properties

Label 7.8.4.2
Base \(\Q_{7}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_8$ (as 8T1)

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Defining polynomial

\( x^{8} + 49 x^{4} - 1029 x^{2} + 12005 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $8$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $8$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.4.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{4} + x^{2} - 3 x + 5 \)
Relative Eisenstein polynomial:$ x^{2} - 7 t \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:\( x^{8} - x^{7} + 27 x^{6} - 28 x^{5} + 151 x^{4} - 350 x^{3} + 500 x^{2} - 846 x + 1157 \)