Properties

Label 3.12.18.82
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(18\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\(x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $18$
Discriminant root field: $\Q_{3}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 3 }) }$: $12$
This field is Galois and abelian over $\Q_{3}.$
Visible slopes:$[2]$

Intermediate fields

$\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.3.4.2, 3.4.2.1, 3.6.8.3, 3.6.9.3, 3.6.9.10

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: \( x^{6} + 6 x^{5} + 6 x^{4} + 3 x^{3} + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$2z^{2} + 1$,$z^{3} + 2$
Associated inertia:$1$,$1$
Indices of inseparability:$[4, 0]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_6$
Wild inertia group:$C_3$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:$[2]$
Galois mean slope:$3/2$
Galois splitting model:$x^{12} - x^{6} + 1$