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

Related objects


Learn more

Defining polynomial

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


Base field: $\Q_{3}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[5/4]$

Intermediate fields

$\Q_{3}(\sqrt{3\cdot 2})$

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

Ramification polygon

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

Invariants of the Galois closure

Galois group:$\SOPlus(4,2)$ (as 6T13)
Inertia group:$C_3^2:C_4$ (as 6T10)
Wild inertia group:$C_3^2$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:$[5/4, 5/4]$
Galois mean slope:$43/36$
Galois splitting model: $x^{6} + 3 x + 3$ Copy content Toggle raw display