Properties

Label 13.13.15.8
Base \(\Q_{13}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(15\)
Galois group $C_{13}:C_4$ (as 13T4)

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Defining polynomial

\(x^{13} + 91 x^{3} + 13\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $13$
Ramification exponent $e$: $13$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
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:$[5/4]$

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: \( x^{13} + 91 x^{3} + 13 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{3} + 5$
Associated inertia:$1$
Indices of inseparability:$[3, 0]$

Invariants of the Galois closure

Galois group:$C_{13}:C_4$ (as 13T4)
Inertia group:$C_{13}:C_4$ (as 13T4)
Wild inertia group:$C_{13}$
Unramified degree:$1$
Tame degree:$4$
Wild slopes:$[5/4]$
Galois mean slope:$63/52$
Galois splitting model:Not computed