Properties

Label 5.13.12.1
Base \(\Q_{5}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(12\)
Galois group $C_{13}:C_4$ (as 13T4)

Related objects

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Defining polynomial

\( x^{13} - 5 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $13$
Ramification exponent $e$ : $13$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{5}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $1$
This field is not Galois over $\Q_{5}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{13} - 5 \)

Invariants of the Galois closure

Galois group:$D_{13}.C_2$ (as 13T4)
Inertia group:$C_{13}$
Unramified degree:$4$
Tame degree:$13$
Wild slopes:None
Galois mean slope:$12/13$
Galois splitting model:Not computed