Properties

Label 53.8.6.1
Base \(\Q_{53}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} + 196 x^{7} + 14414 x^{6} + 471772 x^{5} + 5822555 x^{4} + 953932 x^{3} + 819902 x^{2} + 24880828 x + 304519678\) 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: $\Q_{53}$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 53 }) }$: $8$
This field is Galois and abelian over $\Q_{53}.$
Visible slopes:None

Intermediate fields

$\Q_{53}(\sqrt{2})$, $\Q_{53}(\sqrt{53})$, $\Q_{53}(\sqrt{53\cdot 2})$, 53.4.2.1, 53.4.3.1, 53.4.3.2

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: \( x^{4} + 53 \) $\ \in\Q_{53}(t)[x]$ 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:$C_2\times C_4$ (as 8T2)
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