Properties

Label 13.13.13.8
Base \(\Q_{13}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(13\)
Galois group $F_{13}$ (as 13T6)

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Defining polynomial

\(x^{13} + 65 x + 13\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $13$
Ramification exponent $e$: $13$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{13}(\sqrt{13\cdot 2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 13 }) }$: $1$
This field is not Galois over $\Q_{13}.$
Visible slopes:$[13/12]$

Intermediate fields

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}$
Relative Eisenstein polynomial: \( x^{13} + 65 x + 13 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 8$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$F_{13}$ (as 13T6)
Inertia group:$F_{13}$ (as 13T6)
Wild inertia group:$C_{13}$
Unramified degree:$1$
Tame degree:$12$
Wild slopes:$[13/12]$
Galois mean slope:$167/156$
Galois splitting model:Not computed