Properties

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

Related objects

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Defining polynomial

\( x^{3} - 19 \)

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: $1$
$|\Gal(K/\Q_{ 19 })|$: $3$
This field is Galois and abelian over $\Q_{19}$.

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:\( x^{3} - 19 \)

Invariants of the Galois closure

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