Base \(\Q_{13}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(16\)
Galois group $F_{13}$ (as 13T6)

Related objects

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Defining polynomial

\( x^{13} + 143 x^{4} + 13 \)


Base field: $\Q_{13}$
Degree $d$: $13$
Ramification exponent $e$: $13$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
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:\( x^{13} + 143 x^{4} + 13 \)

Invariants of the Galois closure

Galois group:$F_{13}$ (as 13T6)
Inertia group:$C_{13}:C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:[4/3]
Galois mean slope:$50/39$
Galois splitting model:Not computed