Properties

Label 13.6.5.4
Base \(\Q_{13}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} + 26 \)

Invariants

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

$\Q_{13}(\sqrt{13*})$, 13.3.2.1

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:\( x^{6} + 26 \)

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:$x^{6} - 117 x^{4} - 104 x^{3} + 3276 x^{2} + 6864 x + 1664$