Properties

Label 13.13.14.11
Base \(\Q_{13}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(14\)
Galois group $F_{13}$ (as 13T6)

Related objects

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Defining polynomial

\( x^{13} + 130 x^{2} + 13 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $13$
Ramification exponent $e$ : $13$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{13}(\sqrt{*})$
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:\( x^{13} + 130 x^{2} + 13 \)

Invariants of the Galois closure

Galois group:$F_{13}$ (as 13T6)
Inertia group:$C_{13}:C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:[7/6]
Galois mean slope:$89/78$
Galois splitting model:Not computed