Properties

Label 3.12.14.10
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(14\)
Galois group $C_6\times S_3$ (as 12T18)

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Defining polynomial

\(x^{12} - 6 x^{9} + 12 x^{8} + 51 x^{6} - 36 x^{5} + 36 x^{4} - 18 x^{3} + 36 x^{2} + 9\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{3}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $6$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[3/2]$

Intermediate fields

$\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.2.1, 3.6.7.3

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

Unramified/totally ramified tower

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

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_6\times S_3$ (as 12T18)
Inertia group:Intransitive group isomorphic to $S_3$
Wild inertia group:$C_3$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:$[3/2]$
Galois mean slope:$7/6$
Galois splitting model:$x^{12} - 6 x^{11} + 45 x^{10} - 170 x^{9} + 606 x^{8} - 1470 x^{7} + 3011 x^{6} - 4536 x^{5} + 5490 x^{4} - 4828 x^{3} + 3093 x^{2} - 1236 x + 211$