Properties

Label 13.12.10.6
Base \(\Q_{13}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\( x^{12} - 143 x^{6} + 5408 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
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, 13.4.2.2, 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:$\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{6} - 13 t^{5} \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} + 39 x^{10} - 26 x^{9} - 234 x^{8} + 1872 x^{7} + 14950 x^{6} + 215982 x^{5} + 876603 x^{4} + 1311778 x^{3} + 12435696 x^{2} - 17262336 x + 143309296$