Properties

Label 61.6.3.2
Base \(\Q_{61}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} - 3721 x^{2} + 2269810\)  Toggle raw display

Invariants

Base field: $\Q_{61}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{61}(\sqrt{61\cdot 2})$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 61 }) }$: $6$
This field is Galois and abelian over $\Q_{61}.$

Intermediate fields

$\Q_{61}(\sqrt{61\cdot 2})$, 61.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:61.3.0.1 $\cong \Q_{61}(t)$ where $t$ is a root of \( x^{3} - x + 10 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 61 t \)$\ \in\Q_{61}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{6} - x^{5} + 106 x^{4} - 106 x^{3} + 3256 x^{2} - 3256 x + 26881$