Defining polynomial
| \( x^{12} + 90 x^{11} + 315 x^{10} - 84 x^{9} - 225 x^{8} - 243 x^{7} - 9 x^{6} + 54 x^{5} - 243 x^{4} - 135 x^{3} - 162 x^{2} + 243 x - 162 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $1$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.4.0.1 |
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} - 3 t^{2} + 3 t - 3\right) x^{2} - 3 t^{3} + 3 t^{2} + 3 t \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 12T173 |
| Inertia group: | Intransitive group isomorphic to $C_3^4$ |
| Unramified degree: | $8$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2] |
| Galois mean slope: | $160/81$ |
| Galois splitting model: | $x^{12} - 12 x^{10} - 24 x^{9} - 54 x^{8} - 120 x^{7} + 52 x^{6} + 360 x^{5} + 513 x^{4} + 872 x^{3} + 1656 x^{2} + 1632 x + 496$ |