Properties

Label 3.12.12.30
Base \(\Q_{3}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(12\)
Galois group $PSU(3,2):C_2$ (as 12T84)

Related objects

Learn more about

Defining polynomial

\( x^{12} - 3 x^{10} + 3 x^{7} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} - 3 x - 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{3})$, 3.4.3.2

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:\( x^{12} - 3 x^{10} + 3 x^{7} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} - 3 x - 3 \)

Invariants of the Galois closure

Galois group:$PSU(3,2):C_2$ (as 12T84)
Inertia group:12T46
Unramified degree:$2$
Tame degree:$8$
Wild slopes:[9/8, 9/8]
Galois mean slope:$79/72$
Galois splitting model:$x^{12} - 6 x^{10} - 4 x^{9} + 6 x^{8} + 24 x^{5} + 21 x^{4} + 8 x^{3} + 18 x^{2} + 12 x - 2$