Properties

Label 23.6.4.2
Base \(\Q_{23}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $S_3\times C_3$ (as 6T5)

Related objects

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Defining polynomial

\( x^{6} - 23 x^{3} + 3703 \)

Invariants

Base field: $\Q_{23}$
Degree $d$ : $6$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{23}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 23 })|$: $3$
This field is not Galois over $\Q_{23}$.

Intermediate fields

$\Q_{23}(\sqrt{*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{23}(\sqrt{*})$ $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{2} - x + 7 \)
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 6T5)
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:$x^{6} - 23 x^{3} + 3703$