Properties

Label 13.14.13.2
Base \(\Q_{13}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(13\)
Galois group $D_{14}$ (as 14T3)

Related objects

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Defining polynomial

\( x^{14} + 26 \)

Invariants

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

Intermediate fields

$\Q_{13}(\sqrt{13*})$, 13.7.6.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}$
Relative Eisenstein polynomial:\( x^{14} + 26 \)

Invariants of the Galois closure

Galois group:$D_{14}$ (as 14T3)
Inertia group:$C_{14}$
Unramified degree:$2$
Tame degree:$14$
Wild slopes:None
Galois mean slope:$13/14$
Galois splitting model:Not computed