Base \(\Q_{5}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(12\)
Galois group $C_{13}:C_4$ (as 13T4)

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Defining polynomial

\(x^{13} + 5\) Copy content Toggle raw display


Base field: $\Q_{5}$
Degree $d$: $13$
Ramification exponent $e$: $13$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 5 }) }$: $1$
This field is not Galois over $\Q_{5}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial: \( x^{13} + 5 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{12} + 3z^{11} + 3z^{10} + z^{9} + 2z^{7} + z^{6} + z^{5} + 2z^{4} + z^{2} + 3z + 3$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{13}:C_4$ (as 13T4)
Inertia group:$C_{13}$ (as 13T1)
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$13$
Wild slopes:None
Galois mean slope:$12/13$
Galois splitting model:Not computed