Properties

Label 31.4.2.1
Base \(\Q_{31}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $V_4$ (as 4T2)

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Defining polynomial

\( x^{4} + 713 x^{2} + 138384 \)

Invariants

Base field: $\Q_{31}$
Degree $d$ : $4$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $2$
Discriminant root field: $\Q_{31}$
Root number: $1$
$|\Gal(K/\Q_{ 31 })|$: $4$
This field is Galois and abelian over $\Q_{31}$.

Intermediate fields

$\Q_{31}(\sqrt{*})$, $\Q_{31}(\sqrt{31})$, $\Q_{31}(\sqrt{31*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{31}(\sqrt{*})$ $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{2} - x + 12 \)
Relative Eisenstein polynomial:$ x^{2} - 31 t^{2} \in\Q_{31}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
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:\( x^{4} + 713 x^{2} + 138384 \)