Properties

Label 11.11.14.9
Base \(\Q_{11}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(14\)
Galois group $F_{11}$ (as 11T4)

Related objects

Learn more about

Defining polynomial

\( x^{11} + 88 x^{4} + 11 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $11$
Ramification exponent $e$ : $11$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
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:\( x^{11} + 88 x^{4} + 11 \)

Invariants of the Galois closure

Galois group:$F_{11}$ (as 11T4)
Inertia group:$C_{11}:C_5$
Unramified degree:$2$
Tame degree:$5$
Wild slopes:[7/5]
Galois mean slope:$74/55$
Galois splitting model:Not computed