Properties

Label 7.13.12.1
Base \(\Q_{7}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(12\)
Galois group $F_{13}$ (as 13T6)

Related objects

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Defining polynomial

\( x^{13} - 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $13$
Ramification exponent $e$ : $13$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial:\( x^{13} - 7 \)

Invariants of the Galois closure

Galois group:$F_{13}$ (as 13T6)
Inertia group:$C_{13}$
Unramified degree:$12$
Tame degree:$13$
Wild slopes:None
Galois mean slope:$12/13$
Galois splitting model:$x^{13} - 26 x^{12} + 312 x^{11} - 2288 x^{10} + 11440 x^{9} - 41184 x^{8} + 109824 x^{7} - 219648 x^{6} + 329472 x^{5} - 366080 x^{4} + 292864 x^{3} - 159744 x^{2} + 53248 x - 8199$