Properties

Label 13.13.14.11
Base \(\Q_{13}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(14\)
Galois group $F_{13}$ (as 13T6)

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Defining polynomial

\(x^{13} + 130 x^{2} + 13\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $13$
Ramification exponent $e$: $13$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{13}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 13 }) }$: $1$
This field is not Galois over $\Q_{13}.$
Visible slopes:$[7/6]$

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: \( x^{13} + 130 x^{2} + 13 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 6$
Associated inertia:$2$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$F_{13}$ (as 13T6)
Inertia group:$C_{13}:C_6$ (as 13T5)
Wild inertia group:$C_{13}$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:$[7/6]$
Galois mean slope:$89/78$
Galois splitting model:Not computed