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

Related objects


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Defining polynomial

\(x^{6} + 38\) Copy content Toggle raw display


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

Intermediate fields


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: \( x^{6} + 38 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{5} + 6z^{4} + 15z^{3} + z^{2} + 15z + 6$
Associated inertia:$1$
Indices of inseparability:$[0]$

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:$x^{6} - x^{5} - 55 x^{4} + 160 x^{3} + 246 x^{2} - 1107 x + 729$