Base \(\Q_{47}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} - 2209 x^{2} + 207646\)  Toggle raw display


Base field: $\Q_{47}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{47}(\sqrt{47\cdot 5})$
Root number: $-i$
$|\Gal(K/\Q_{ 47 })|$: $6$
This field is Galois and abelian over $\Q_{47}.$

Intermediate fields

$\Q_{47}(\sqrt{47\cdot 5})$,

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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{47}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 47 t \)$\ \in\Q_{47}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed