Properties

Label 13.12.8.2
Base \(\Q_{13}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\( x^{12} + 169 x^{6} - 2197 x^{3} + 57122 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{13}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $12$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{*})$, 13.3.2.1, 13.4.0.1, 13.6.4.1

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

Unramified/totally ramified tower

Unramified subfield:13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{4} + x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} - 13 t \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{12} + 39 x^{10} - 26 x^{9} + 1521 x^{8} + 3042 x^{7} + 59995 x^{6} + 79092 x^{5} + 2260713 x^{4} + 1524718 x^{3} + 1028196 x^{2} + 685464 x + 456976$