Defining polynomial
| \( x^{12} + 306 x^{11} + 135 x^{10} + 156 x^{9} + 162 x^{8} - 135 x^{7} + 117 x^{6} + 162 x^{3} - 162 \) |
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, 3.6.10.4 |
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\right) x^{2} + \left(9 t^{3} - 9 t^{2}\right) x + 3 t^{3} + 3 t^{2} + 3 t - 6 \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: | $x^{12} - 72 x^{6} + 648$ |