Base \(\Q_{13}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $C_8:C_2$ (as 8T7)

Related objects

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Defining polynomial

\(x^{8} - 52\)  Toggle raw display


Base field: $\Q_{13}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{13}(\sqrt{13})$
Root number: $1$
$|\Aut(K/\Q_{ 13 })|$: $4$
This field is not Galois 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:\( x^{8} - 52 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$OD_{16}$ (as 8T7)
Inertia group:$C_8$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:$x^{8} + 26 x^{6} + 65 x^{4} + 52 x^{2} + 13$