Defining polynomial
| \( x^{12} + 36 x^{11} + 111 x^{10} + 90 x^{9} + 36 x^{8} + 90 x^{7} + 30 x^{6} + 108 x^{5} - 36 x^{4} + 54 x^{3} - 81 x^{2} + 54 x - 18 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $12$ |
| This field is Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.3.4.4 x3, 3.4.2.1, 3.6.8.5, 3.6.9.4 x3, 3.6.9.12 x3 |
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} + \left(9 t + 9\right) x^{5} + \left(9 t - 3\right) x^{4} + \left(-9 t - 9\right) x^{2} - 9 x - 12 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 12T3) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | [2] |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{12} - 2 x^{9} + 2 x^{6} + 2 x^{3} + 1$ |