Properties

Label 19.12.0.1
Base \(\Q_{19}\)
Degree \(12\)
e \(1\)
f \(12\)
c \(0\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\(x^{12} + 3 x^{7} + 2 x^{6} + 18 x^{5} + 2 x^{4} + 9 x^{3} + 16 x^{2} + 7 x + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $12$
Ramification exponent $e$: $1$
Residue field degree $f$: $12$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{19}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $12$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{2})$, 19.3.0.1, 19.4.0.1, 19.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.12.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{12} + 3 x^{7} + 2 x^{6} + 18 x^{5} + 2 x^{4} + 9 x^{3} + 16 x^{2} + 7 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 19 \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$12$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$