Properties

Label 13.12.6.1
Base \(\Q_{13}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\( x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{13}$
Root number: $-1$
$|\Gal(K/\Q_{ 13 })|$: $12$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{*})$, $\Q_{13}(\sqrt{13})$, $\Q_{13}(\sqrt{13*})$, 13.3.0.1, 13.4.2.1, 13.6.0.1, 13.6.3.1, 13.6.3.2

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

Unramified/totally ramified tower

Unramified subfield:13.6.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{6} + x^{2} - 2 x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 13 t^{2} \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{12} - 39 x^{10} - 4 x^{9} + 471 x^{8} + 36 x^{7} - 2247 x^{6} - 240 x^{5} + 4269 x^{4} + 812 x^{3} - 3111 x^{2} - 738 x + 521$