Properties

Label 53.8.6.3
Base \(\Q_{53}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_8$ (as 8T1)

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Defining polynomial

\(x^{8} - 31588 x^{4} - 79924477\) Copy content Toggle raw display

Invariants

Base field: $\Q_{53}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{53}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 53 }) }$: $8$
This field is Galois and abelian over $\Q_{53}.$
Visible slopes:None

Intermediate fields

$\Q_{53}(\sqrt{2})$, 53.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{53}(\sqrt{2})$ $\cong \Q_{53}(t)$ where $t$ is a root of \( x^{2} + 49 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 742 t + 2385 \) $\ \in\Q_{53}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{3} + 4z^{2} + 6z + 4$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:Not computed