Properties

Label 3.12.12.21
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(12\)
Galois group $C_3^2:F_9$ (as 12T173)

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Defining polynomial

\(x^{12} - 6 x^{11} + 60 x^{10} - 114 x^{9} + 2286 x^{8} + 33750 x^{7} + 101790 x^{6} + 91152 x^{5} + 24867 x^{4} + 6102 x^{3} + 5832 x^{2} - 1296 x + 81\) 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: \( x^{3} + \left(6 t^{3} + 6 t^{2} + 3 t + 6\right) x^{2} + \left(6 t^{3} + 6 t + 3\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t^{3} + t + 2$
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: $x^{12} - 12 x^{10} - 8 x^{9} + 54 x^{8} + 72 x^{7} - 203 x^{6} - 216 x^{5} + 651 x^{4} + 252 x^{3} - 855 x^{2} - 108 x + 373$ Copy content Toggle raw display