Properties

Label 3.6.10.11
Base \(\Q_{3}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(10\)
Galois group $C_3^2:D_4$ (as 6T13)

Related objects

Learn more about

Defining polynomial

\( x^{6} + 3 x^{5} + 15 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $6$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
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:\( x^{6} + 3 x^{5} + 15 \)

Invariants of the Galois closure

Galois group:$S_3\wr C_2$ (as 6T13)
Inertia group:$C_3^2:C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[9/4, 9/4]
Galois mean slope:$25/12$
Galois splitting model:$x^{6} - 6 x^{4} - 2 x^{3} + 9 x^{2} + 6 x - 2$