Properties

Label 13.14.12.2
Base \(\Q_{13}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(12\)
Galois group $C_7 \wr C_2$ (as 14T8)

Related objects

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Defining polynomial

\( x^{14} - 13 x^{7} + 338 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $14$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{13}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 13 })|$: $7$
This field is not Galois over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{7} - 13 t \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_7\times D_7$ (as 14T8)
Inertia group:Intransitive group isomorphic to $C_7$
Unramified degree:$14$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed