Properties

Label 3.12.16.25
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(16\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\(x^{12} + 93 x^{11} - 36 x^{10} + 357 x^{9} + 270 x^{8} + 324 x^{7} + 207 x^{6} - 216 x^{5} - 324 x^{4} - 54 x^{3} - 81 x^{2} - 324\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $16$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $12$
This field is Galois and abelian over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 3.3.4.3, 3.4.0.1, 3.6.8.4

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 \( x^{4} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{3} + \left(-3 t^{2} - 3 t - 3\right) x^{2} - 3 t^{3} + 3 t^{2} + 3 \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$4/3$
Galois splitting model:$x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$