Defining polynomial
|
\(x^{12} + 36 x^{7} - 12 x^{6} + 81 x^{4} + 486 x^{2} - 216 x + 117\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $6$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.2.1, 3.6.11.21 |
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
\( x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + \left(9 t + 9\right) x^{2} + 18 x + 9 t + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $2z^{2} + 1$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_6\times S_3$ (as 12T18) |
| Inertia group: | Intransitive group isomorphic to $C_3\times S_3$ |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | $[2, 5/2]$ |
| Galois mean slope: | $13/6$ |
| Galois splitting model: | $x^{12} - 30 x^{9} + 288 x^{6} - 378 x^{3} + 567$ |