Properties

Label 19.3.2.2
Base \(\Q_{19}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(2\)
Galois group $C_3$ (as 3T1)

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Defining polynomial

\(x^{3} + 19\) Copy content Toggle raw display

Invariants

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

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial: \( x^{3} + 19 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3$ (as 3T1)
Inertia group:$C_3$ (as 3T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{3} - x^{2} - 6 x + 7$