Properties

Label 3.6.9.15
Base \(\Q_{3}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(9\)
Galois group $S_3\times C_3$ (as 6T5)

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Defining polynomial

\( x^{6} + 6 x^{4} + 6 x^{3} + 12 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $6$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{3}(\sqrt{3*})$
Root number: $-i$
$|\Aut(K/\Q_{ 3 })|$: $3$
This field is not Galois over $\Q_{3}$.

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{6} + 6 x^{4} + 6 x^{3} + 12 \)

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 6T5)
Inertia group:$S_3\times C_3$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:[3/2, 2]
Galois mean slope:$31/18$
Galois splitting model:$x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 12$