Defining polynomial
| \( x^{12} + 105 x^{11} - 513 x^{10} - 834 x^{9} - 117 x^{8} - 459 x^{7} - 1008 x^{6} - 81 x^{5} - 270 x^{3} + 648 x^{2} - 486 x + 810 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $6$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.4.0.1, 3.6.8.9 |
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(-9 t^{3} - 9 t^{2} - 9 t - 3\right) x^{2} + \left(9 t^{3} - 9 t + 9\right) x - 12 t^{3} + 3 t^{2} + 12 t + 9 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times C_3:C_4$ (as 12T19) |
| Inertia group: | Intransitive group isomorphic to $C_3^2$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2] |
| Galois mean slope: | $16/9$ |
| Galois splitting model: | $x^{12} + 30 x^{10} - 40 x^{9} + 468 x^{8} - 1335 x^{7} + 4738 x^{6} - 14841 x^{5} + 27090 x^{4} - 67130 x^{3} + 51498 x^{2} + 65265 x + 29981$ |