Properties

Label 19.6.5.6
Base \(\Q_{19}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + 95\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{19}(\sqrt{19\cdot 2})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 19 }) }$: $6$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{19\cdot 2})$, 19.3.2.3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial: \( x^{6} + 95 \) Copy content Toggle raw display

Ramification polygon

Not computed

Invariants of the Galois closure

Galois group: $C_6$ (as 6T1)
Inertia group: $C_6$ (as 6T1)
Wild inertia group: $C_1$
Unramified degree: $1$
Tame degree: $6$
Wild slopes: None
Galois mean slope: $5/6$
Galois splitting model:$x^{6} - x^{5} + 2 x^{4} + 122 x^{3} + 474 x^{2} + 888 x + 881$