Properties

Label 19.4.2.2
Base \(\Q_{19}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_4$ (as 4T1)

Related objects

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Defining polynomial

\( x^{4} - 19 x^{2} + 722 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $4$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $2$
Discriminant root field: $\Q_{19}(\sqrt{*})$
Root number: $-1$
$|\Gal(K/\Q_{ 19 })|$: $4$
This field is Galois and abelian over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}(\sqrt{*})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 19 t \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$C_4$ (as 4T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{4} - x^{3} - 63 x^{2} - 61 x + 263$