Properties

Label 23.12.11.2
Base \(\Q_{23}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_{12}$ (as 12T12)

Related objects

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Defining polynomial

\( x^{12} - 23 \)

Invariants

Base field: $\Q_{23}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $11$
Discriminant root field: $\Q_{23}(\sqrt{23*})$
Root number: $-i$
$|\Aut(K/\Q_{ 23 })|$: $2$
This field is not Galois over $\Q_{23}$.

Intermediate fields

$\Q_{23}(\sqrt{23})$, 23.3.2.1, 23.4.3.2, 23.6.5.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{23}$
Relative Eisenstein polynomial:\( x^{12} - 23 \)

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:Not computed