Properties

Label 3.6.9.5
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

\(x^{6} + 3 x^{5} + 3 x^{4} + 6\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $3$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[2]$

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: \( x^{6} + 3 x^{5} + 3 x^{4} + 6 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$2z^{2} + 1$,$z^{3} + 2$
Associated inertia:$1$,$1$
Indices of inseparability:$[4, 0]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 6T5)
Inertia group:$C_3\times S_3$ (as 6T5)
Wild inertia group:$C_3^2$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:$[3/2, 2]$
Galois mean slope:$31/18$
Galois splitting model:$x^{6} + 84 x^{4} - 84 x^{3} + 1764 x^{2} - 3528 x + 7791$