Properties

Label 23.9.6.1
Base \(\Q_{23}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $S_3\times C_3$ (as 9T4)

Related objects

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Defining polynomial

\( x^{9} - 529 x^{3} + 48668 \)

Invariants

Base field: $\Q_{23}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{23}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 23 })|$: $3$
This field is not Galois over $\Q_{23}.$

Intermediate fields

23.3.2.1, 23.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:23.3.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{3} - x + 4 \)
Relative Eisenstein polynomial:$ x^{3} - 23 t \in\Q_{23}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 9T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed