Properties

Label 3.6.8.10
Base \(\Q_{3}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(8\)
Galois group $S_3\times C_3$ (as 6T5)

Related objects

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Defining polynomial

\( x^{6} + 6 x^{5} + 36 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $6$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $3$
This field is not Galois over $\Q_{3}$.

Intermediate fields

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

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} + 3 x^{2} + 6 t + 6 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 6T5)
Inertia group:Intransitive group isomorphic to $C_3^2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$16/9$
Galois splitting model:$x^{6} - 14 x^{3} + 63 x^{2} - 84 x + 77$