Properties

Label 19.6.4.3
Base \(\Q_{19}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

$( x^{2} + 18 x + 2 )^{3} + 190 ( x^{2} + 18 x + 2 )^{2} + \left(38 x + 39387\right) ( x^{2} + 18 x + 2 ) + 15618 x + 140923$ Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{19}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $6$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{2})$, 19.3.2.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} + 18 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 95 x + 19 \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Not computed

Invariants of the Galois closure

Galois group: $C_6$ (as 6T1)
Inertia group: Intransitive group isomorphic to $C_3$
Wild inertia group: $C_1$
Unramified degree: $2$
Tame degree: $3$
Wild slopes: None
Galois mean slope: $2/3$
Galois splitting model: $x^{6} - x^{5} + 20 x^{4} - 23 x^{3} + 186 x^{2} - 168 x + 392$ Copy content Toggle raw display