Defining polynomial
| \( x^{12} + 42 x^{11} - 57 x^{10} - 36 x^{9} - 117 x^{8} - 54 x^{7} - 18 x^{6} - 27 x^{5} + 54 x^{4} + 108 x^{3} - 81 x^{2} + 81 x + 81 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.4.0.1 |
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(3 t^{3} + 3 t^{2}\right) x^{2} + \left(3 t^{3} + 3 t^{2} - 3\right) x - 3 t - 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 12T173 |
| Inertia group: | Intransitive group isomorphic to $C_3:(C_3^3:C_2)$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 3/2, 3/2, 3/2] |
| Galois mean slope: | $241/162$ |
| Galois splitting model: | $x^{12} + 12 x^{10} - 2 x^{9} + 54 x^{8} - 18 x^{7} + 33 x^{6} - 54 x^{5} - 369 x^{4} - 250 x^{3} - 675 x^{2} - 588 x - 137$ |