Base \(\Q_{13}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $C_6$ (as 6T1)

Related objects

Learn more about

Defining polynomial

\(x^{6} - 13\)  Toggle raw display


Base field: $\Q_{13}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{13}(\sqrt{13})$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $6$
This field is Galois and abelian over $\Q_{13}.$

Intermediate fields


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:\( x^{6} - 13 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:$C_6$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$