Properties

Label 19.13.0.1
Base \(\Q_{19}\)
Degree \(13\)
e \(1\)
f \(13\)
c \(0\)
Galois group $C_{13}$

Related objects

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Defining polynomial

\( x^{13} - x + 4 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $13$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $13$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{19}$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $13$
This field is Galois and abelian over $\Q_{19}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$.

Unramified/totally ramified tower

Unramified subfield:19.13.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{13} - x + 4 \)
Relative Eisenstein polynomial:$ x - 19 \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{13}$
Inertia group:Trivial
Unramified degree:$13$
Tame degree:$1$
Wild slopes:None
Galois Mean Slope:$0$
Global Splitting Model:\( x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} - 215 x^{4} - 291 x^{3} + 68 x^{2} + 10 x - 1 \)