Properties

Label 11.12.11.2
Base \(\Q_{11}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_{12}$ (as 12T12)

Related objects

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Defining polynomial

\(x^{12} - 11\)  Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{11}(\sqrt{11\cdot 2})$
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, 11.4.3.2, 11.6.5.1

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:\( x^{12} - 11 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$D_{12}$ (as 12T12)
Inertia group:$C_{12}$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:$x^{12} - 22 x^{8} + 209 x^{4} - 44$