Defining polynomial
|
\(x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $4$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.3.2, 3.4.0.1, 3.6.6.3 |
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
\( x^{4} + 2 x^{3} + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + \left(3 t + 3\right) x^{2} + 3 x + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_4\times S_3$ (as 12T11) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2]$ |
| Galois mean slope: | $7/6$ |
| Galois splitting model: | $x^{12} - 5 x^{9} + 15 x^{6} - 15 x^{3} + 5$ |