Properties

Label 13.7.6.1
Base \(\Q_{13}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $D_{7}$ (as 7T2)

Related objects

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Defining polynomial

\( x^{7} - 13 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $7$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $6$
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:\( x^{7} - 13 \)

Invariants of the Galois closure

Galois group:$D_7$ (as 7T2)
Inertia group:$C_7$
Unramified degree:$2$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:$x^{7} - 3 x^{6} + 2 x^{5} + 6 x^{4} + 3 x^{3} - 16 x^{2} - 7 x - 7$