Properties

Label 23.12.10.4
Base \(\Q_{23}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

Related objects

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Defining polynomial

\( x^{12} - 23 x^{6} + 3703 \)

Invariants

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

Intermediate fields

$\Q_{23}(\sqrt{*})$, 23.4.2.2, 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 \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 $C_6$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:Not computed