Defining polynomial
| \( x^{12} + 18 x^{11} + 54 x^{10} + 18 x^{9} - 81 x^{8} - 54 x^{7} + 90 x^{6} + 81 x^{5} + 81 x^{4} - 27 x^{3} + 81 \) |
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: | $-1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $2$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.4.0.1, 3.6.10.5 |
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} + 9 t^{2} x^{2} + \left(9 t^{3} + 9 t - 9\right) x - 12 t^{3} - 6 t^{2} - 3 t - 12 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2.S_3^2$ (as 12T39) |
| Inertia group: | Intransitive group isomorphic to $C_3:S_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 5/2] |
| Galois mean slope: | $37/18$ |
| Galois splitting model: | $x^{12} - 36 x^{6} + 162$ |