Properties

Label 23.12.8.2
Base \(\Q_{23}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

Related objects

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Defining polynomial

\( x^{12} - 12167 x^{3} + 3078251 \)

Invariants

Base field: $\Q_{23}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{23}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 23 })|$: $6$
This field is not Galois over $\Q_{23}.$

Intermediate fields

$\Q_{23}(\sqrt{*})$, 23.4.0.1, 23.6.4.2

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

Unramified/totally ramified tower

Unramified subfield:23.4.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{4} - x + 11 \)
Relative Eisenstein polynomial:$ x^{3} - 23 t \in\Q_{23}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times C_3:C_4$ (as 12T19)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$12$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed