Properties

Label 3.6.10.1
Base \(\Q_{3}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(10\)
Galois group $D_{6}$ (as 6T3)

Related objects

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Defining polynomial

\( x^{6} - 18 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $6$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.3.5.3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} + 6 t + 24 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$D_6$ (as 6T3)
Inertia group:Intransitive group isomorphic to $S_3$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:[5/2]
Galois mean slope:$11/6$
Galois splitting model:$x^{6} - 18$