Properties

Label 137.10.8.1
Base \(\Q_{137}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $F_5$ (as 10T4)

Related objects

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Defining polynomial

\(x^{10} - 137 x^{5} + 112614\)  Toggle raw display

Invariants

Base field: $\Q_{137}$
Degree $d$: $10$
Ramification exponent $e$: $5$
Residue field degree $f$: $2$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{137}(\sqrt{3})$
Root number: $1$
$|\Aut(K/\Q_{ 137 })|$: $2$
This field is not Galois over $\Q_{137}.$

Intermediate fields

$\Q_{137}(\sqrt{3})$, 137.5.4.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{137}(\sqrt{3})$ $\cong \Q_{137}(t)$ where $t$ is a root of \( x^{2} - x + 6 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{5} - 137 t \)$\ \in\Q_{137}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$F_5$ (as 10T4)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed