Properties

Label 11.13.12.1
Base \(\Q_{11}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(12\)
Galois group $F_{13}$ (as 13T6)

Related objects

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Defining polynomial

\( x^{13} - 11 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $13$
Ramification exponent $e$ : $13$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{11}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 11 })|$: $1$
This field is not Galois over $\Q_{11}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{13} - 11 \)

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:$x^{13} - 39 x^{12} + 702 x^{11} - 7722 x^{10} + 57915 x^{9} - 312741 x^{8} + 1250964 x^{7} - 3752892 x^{6} + 8444007 x^{5} - 14073345 x^{4} + 16888014 x^{3} - 13817466 x^{2} + 6908733 x - 1594334$