Properties

Label 3.6.11.7
Base \(\Q_{3}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(11\)
Galois group $S_3$ (as 6T2)

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Defining polynomial

\(x^{6} + 9 x^{2} + 12\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 3 }) }$: $6$
This field is Galois over $\Q_{3}.$
Visible slopes:$[5/2]$

Intermediate fields

$\Q_{3}(\sqrt{3\cdot 2})$, 3.3.5.2 x3

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: \( x^{6} + 9 x^{2} + 12 \) Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $S_3$ (as 6T2)
Inertia group: $S_3$ (as 6T2)
Wild inertia group: $C_3$
Unramified degree: $1$
Tame degree: $2$
Wild slopes: $[5/2]$
Galois mean slope: $11/6$
Galois splitting model:$x^{6} - 3 x^{5} + 6 x^{4} + 35 x^{3} - 57 x^{2} - 66 x + 484$