Properties

Label 53.4.2.2
Base \(\Q_{53}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_4$ (as 4T1)

Related objects

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Defining polynomial

\( x^{4} - 53 x^{2} + 14045 \)

Invariants

Base field: $\Q_{53}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{53}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 53 })|$: $4$
This field is Galois and abelian over $\Q_{53}.$

Intermediate fields

$\Q_{53}(\sqrt{*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{53}(\sqrt{*})$ $\cong \Q_{53}(t)$ where $t$ is a root of \( x^{2} - x + 5 \)
Relative Eisenstein polynomial:$ x^{2} - 53 t \in\Q_{53}(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:$x^{4} - x^{3} + 66 x^{2} - 66 x + 911$