Properties

Label 3.12.12.14
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(12\)
Galois group $C_3^2:F_9$ (as 12T173)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{12} + 54 x^{10} - 240 x^{9} - 1935 x^{8} + 7236 x^{7} + 45198 x^{6} + 60156 x^{5} + 16686 x^{4} - 13824 x^{3} + 5184 x^{2} - 972 x + 81\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{3}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[3/2]$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 3.4.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} + 2 x^{3} + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + \left(3 t^{3} + 6 t^{2} + 6 t + 3\right) x^{2} + \left(6 t^{3} + 6 t^{2} + 6 t\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3^2:F_9$ (as 12T173)
Inertia group:Intransitive group isomorphic to $C_3^3:S_3$
Wild inertia group:$C_3^4$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:$[3/2, 3/2, 3/2, 3/2]$
Galois mean slope:$241/162$
Galois splitting model: $x^{12} + 12 x^{10} - 15 x^{9} + 54 x^{8} - 135 x^{7} + 169 x^{6} - 405 x^{5} + 447 x^{4} - 481 x^{3} + 549 x^{2} - 228 x + 16$ Copy content Toggle raw display