Properties

Label 23.12.10.2
Base \(\Q_{23}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_6\times S_3$ (as 12T18)

Related objects

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Defining polynomial

\( x^{12} + 299 x^{6} + 25921 \)

Invariants

Base field: $\Q_{23}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{23}$
Root number: $1$
$|\Aut(K/\Q_{ 23 })|$: $6$
This field is not Galois over $\Q_{23}$.

Intermediate fields

$\Q_{23}(\sqrt{*})$, $\Q_{23}(\sqrt{23})$, $\Q_{23}(\sqrt{23*})$, 23.4.2.1, 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:$\Q_{23}(\sqrt{*})$ $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{2} - x + 7 \)
Relative Eisenstein polynomial:$ x^{6} - 23 t^{2} \in\Q_{23}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6\times S_3$ (as 12T18)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} + 299 x^{6} + 25921$