Properties

Label 11.6.5.1
Base \(\Q_{11}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $D_{6}$ (as 6T3)

Related objects

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Defining polynomial

\( x^{6} - 11 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $6$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $5$
Discriminant root field: $\Q_{11}(\sqrt{11})$
Root number: $-i$
$|\Aut(K/\Q_{ 11 })|$: $2$
This field is not Galois over $\Q_{11}$.

Intermediate fields

$\Q_{11}(\sqrt{11})$, 11.3.2.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{6} - 11 \)

Invariants of the Galois closure

Galois group:$D_6$ (as 6T3)
Inertia group:$C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:\( x^{6} - 11 \)