Properties

Label 13.12.11.3
Base \(\Q_{13}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\( x^{12} - 208 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $11$
Discriminant root field: $\Q_{13}(\sqrt{13})$
Root number: $-1$
$|\Gal(K/\Q_{ 13 })|$: $12$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{13})$, 13.3.2.1, 13.4.3.1, 13.6.5.3

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:\( x^{12} - 208 \)

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:$C_{12}$
Unramified degree:$1$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:$x^{12} + 78 x^{10} - 104 x^{9} + 1638 x^{8} - 4992 x^{7} + 3393 x^{6} + 53937 x^{5} + 39702 x^{4} - 24908 x^{3} - 426465 x^{2} + 2067 x + 1394549$