Properties

Label 13.4.2.2
Base \(\Q_{13}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_4$ (as 4T1)

Related objects

Learn more about

Defining polynomial

\( x^{4} - 13 x^{2} + 338 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $4$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $2$
Discriminant root field: $\Q_{13}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $4$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 13 t \in\Q_{13}(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:$x^{4} - x^{3} + 16 x^{2} - 16 x + 61$