Properties

Label 13.10.9.2
Base \(\Q_{13}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $F_{5}\times C_2$ (as 10T5)

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Defining polynomial

\(x^{10} + 26\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{13}(\sqrt{13\cdot 2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 13 }) }$: $2$
This field is not Galois over $\Q_{13}.$
Visible slopes:None

Intermediate fields

$\Q_{13}(\sqrt{13\cdot 2})$, 13.5.4.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}$
Relative Eisenstein polynomial: \( x^{10} + 26 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{9} + 10z^{8} + 6z^{7} + 3z^{6} + 2z^{5} + 5z^{4} + 2z^{3} + 3z^{2} + 6z + 10$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:$C_{10}$ (as 10T1)
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:$x^{10} + 26$