Properties

Label 23.13.0.1
Base \(\Q_{23}\)
Degree \(13\)
e \(1\)
f \(13\)
c \(0\)
Galois group $C_{13}$ (as 13T1)

Related objects

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Defining polynomial

\( x^{13} - 5 x + 3 \)

Invariants

Base field: $\Q_{23}$
Degree $d$ : $13$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $13$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{23}$
Root number: $1$
$|\Gal(K/\Q_{ 23 })|$: $13$
This field is Galois and abelian over $\Q_{23}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$.

Unramified/totally ramified tower

Unramified subfield:23.13.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{13} - 5 x + 3 \)
Relative Eisenstein polynomial:$ x - 23 \in\Q_{23}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{13}$ (as 13T1)
Inertia group:Trivial
Unramified degree:$13$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{13} - x^{12} - 60 x^{11} + 27 x^{10} + 1199 x^{9} - 33 x^{8} - 9610 x^{7} - 3352 x^{6} + 33548 x^{5} + 20328 x^{4} - 47723 x^{3} - 34869 x^{2} + 21271 x + 15667$