Defining polynomial
| \( x^{12} + 18 x^{11} + 27 x^{10} + 27 x^{9} - 36 x^{8} + 36 x^{7} - 33 x^{6} + 27 x^{4} - 36 x^{3} - 27 x^{2} - 27 x + 18 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $22$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $2$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.4.2.2, 3.6.10.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{6} + 9 t x^{5} + \left(9 t + 9\right) x^{4} + \left(6 t - 3\right) x^{3} + \left(9 t - 9\right) x^{2} - 9 x + 6 t + 3 \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: | $x^{12} - 24 x^{6} + 72$ |