Properties

Label 139.12.0.1
Base \(\Q_{139}\)
Degree \(12\)
e \(1\)
f \(12\)
c \(0\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\( x^{12} - x + 22 \)

Invariants

Base field: $\Q_{139}$
Degree $d$: $12$
Ramification exponent $e$: $1$
Residue field degree $f$: $12$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{139}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 139 })|$: $12$
This field is Galois and abelian over $\Q_{139}.$

Intermediate fields

$\Q_{139}(\sqrt{2})$, 139.3.0.1, 139.4.0.1, 139.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:139.12.0.1 $\cong \Q_{139}(t)$ where $t$ is a root of \( x^{12} - x + 22 \)
Relative Eisenstein polynomial:$ x - 139 \in\Q_{139}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:trivial
Unramified degree:$12$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{12} - x^{11} + 3 x^{10} - 11 x^{9} - 17 x^{8} + 169 x^{7} + 325 x^{6} - 167 x^{5} - 804 x^{4} - 160 x^{3} + 1102 x^{2} + 780 x + 1179$