Properties

Label 13.8.4.1
Base \(\Q_{13}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\( x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $8$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{13}$
Root number: $-1$
$|\Gal(K/\Q_{ 13 })|$: $8$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{*})$, $\Q_{13}(\sqrt{13})$, $\Q_{13}(\sqrt{13*})$, 13.4.0.1, 13.4.2.1, 13.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{4} + x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 13 t^{2} \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
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:$x^{8} - x^{7} + 4 x^{6} - 7 x^{5} + 19 x^{4} + 21 x^{3} + 36 x^{2} + 27 x + 81$