Properties

Label 3.9.13.3
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(13\)
Galois group $C_3 \wr S_3 $ (as 9T20)

Related objects

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Defining polynomial

\( x^{9} + 3 x^{6} + 6 x^{5} + 6 x^{3} + 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $9$
Ramification exponent $e$ : $9$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $13$
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

3.3.3.1

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:\( x^{9} + 3 x^{6} + 6 x^{5} + 6 x^{3} + 3 \)

Invariants of the Galois closure

Galois group:$C_3\wr S_3$ (as 9T20)
Inertia group:$(C_3^2:C_3):C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:[3/2, 3/2, 5/3]
Galois mean slope:$85/54$
Galois splitting model:$x^{9} - 3 x^{7} - 6 x^{6} - 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} + 5$