Properties

Label 13.11.10.1
Base \(\Q_{13}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(10\)
Galois group $F_{11}$ (as 11T4)

Related objects

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Defining polynomial

\( x^{11} - 13 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $11$
Ramification exponent $e$ : $11$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $10$
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:\( x^{11} - 13 \)

Invariants of the Galois closure

Galois group:$F_{11}$ (as 11T4)
Inertia group:$C_{11}$
Unramified degree:$10$
Tame degree:$11$
Wild slopes:None
Galois mean slope:$10/11$
Galois splitting model:$x^{11} - 13$