Defining polynomial
| \( x^{12} + 102 x^{11} + 207 x^{10} + 192 x^{9} + 108 x^{8} - 216 x^{7} - 306 x^{6} - 189 x^{5} + 162 x^{4} - 27 x^{3} - 324 x^{2} - 324 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $3$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 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(12 t^{3} + 6 t^{2} + 6 t - 12\right) x^{2} + \left(-9 t^{3} - 9 t^{2} - 9 t\right) x - 9 t^{3} - 12 t^{2} - 3 t + 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times C_3:S_3.C_2$ (as 12T73) |
| Inertia group: | Intransitive group isomorphic to $C_3^3$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2] |
| Galois mean slope: | $52/27$ |
| Galois splitting model: | $x^{12} + 12 x^{10} - 12 x^{9} + 54 x^{8} - 108 x^{7} + 230 x^{6} - 324 x^{5} + 813 x^{4} - 696 x^{3} + 1098 x^{2} - 1116 x + 1057$ |