Defining polynomial
| \( x^{12} + 324 x^{11} + 207 x^{10} - 24 x^{9} + 297 x^{8} - 324 x^{7} + 270 x^{6} - 81 x^{5} + 81 x^{4} + 243 x^{3} - 243 x - 324 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $4$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.3.5.1, 3.4.0.1, 3.6.10.2 |
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^{2} + 9 t - 9\right) x^{2} + \left(9 t^{2} + 9 t + 9\right) x - 9 t^{3} - 3 t^{2} - 9 t + 9 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_4\times S_3$ (as 12T11) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | [5/2] |
| Galois mean slope: | $11/6$ |
| Galois splitting model: | $x^{12} - 12 x^{10} - 12 x^{9} + 54 x^{8} - 58 x^{6} + 432 x^{4} - 300 x^{3} + 924 x^{2} - 216 x - 271$ |