Defining polynomial
|
\(x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81\)
|
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: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.4.2, 3.4.0.1, 3.6.8.3 |
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
\( x^{4} - x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + \left(-3 t^{3} + 12 t^{2} + 12 t - 3\right) x^{2} + \left(-9 t^{3} - 9 t + 9\right) x - 3 t^{3} + 12 t^{2} + 3 t - 6 \)
$\ \in\Q_{3}(t)[x]$
|
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2]$ |
| Galois mean slope: | $4/3$ |
| Galois splitting model: | $x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1$ |