Properties

Label 23.12.6.1
Base \(\Q_{23}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\(x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628\) Copy content Toggle raw display

Invariants

Base field: $\Q_{23}$
Degree $d$: $12$
Ramification exponent $e$: $2$
Residue field degree $f$: $6$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{23}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 23 }) }$: $12$
This field is Galois and abelian over $\Q_{23}.$
Visible slopes:None

Intermediate fields

$\Q_{23}(\sqrt{5})$, $\Q_{23}(\sqrt{23})$, $\Q_{23}(\sqrt{23\cdot 5})$, 23.3.0.1, 23.4.2.1, 23.6.0.1, 23.6.3.1, 23.6.3.2

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

Unramified/totally ramified tower

Unramified subfield:23.6.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 23 \) $\ \in\Q_{23}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed