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)

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Defining polynomial

\(x^{12} - 138 x^{6} - 217948\) Copy content Toggle raw display

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: $1$
$\card{ \Aut(K/\Q_{ 23 }) }$: $6$
This field is not Galois over $\Q_{23}.$
Visible slopes:None

Intermediate fields

$\Q_{23}(\sqrt{5})$, $\Q_{23}(\sqrt{23})$, $\Q_{23}(\sqrt{23\cdot 5})$, 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:$\Q_{23}(\sqrt{5})$ $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{2} + 21 x + 5 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{6} + 46 t + 414 \) $\ \in\Q_{23}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_6\times S_3$ (as 12T18)
Inertia group:Intransitive group isomorphic to $C_6$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model: $x^{12} + 299 x^{6} + 25921$ Copy content Toggle raw display