Properties

Label 13.13.15.7
Base \(\Q_{13}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(15\)
Galois group $C_{13}:C_4$ (as 13T4)

Related objects

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Defining polynomial

\( x^{13} + 78 x^{3} + 13 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $13$
Ramification exponent $e$ : $13$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $15$
Discriminant root field: $\Q_{13}(\sqrt{13*})$
Root number: $1$
$|\Aut(K/\Q_{ 13 })|$: $1$
This field is not Galois over $\Q_{13}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}$
Relative Eisenstein polynomial:\( x^{13} + 78 x^{3} + 13 \)

Invariants of the Galois closure

Galois group:$D_{13}.C_2$ (as 13T4)
Inertia group:$C_{13}:C_4$
Unramified degree:$1$
Tame degree:$4$
Wild slopes:[5/4]
Galois mean slope:$63/52$
Galois splitting model:Not computed