Properties

Label 13.13.0.1
Base \(\Q_{13}\)
Degree \(13\)
e \(1\)
f \(13\)
c \(0\)
Galois group $C_{13}$ (as 13T1)

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Defining polynomial

\(x^{13} + 12 x + 11\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $13$
Ramification exponent $e$: $1$
Residue field degree $f$: $13$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{13}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 13 }) }$: $13$
This field is Galois and abelian over $\Q_{13}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$.

Unramified/totally ramified tower

Unramified subfield:13.13.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{13} + 12 x + 11 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 13 \) $\ \in\Q_{13}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_{13}$ (as 13T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$13$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} - 215 x^{4} - 291 x^{3} + 68 x^{2} + 10 x - 1$