Defining polynomial
| \( x^{12} + 108 x^{6} - 243 x^{2} + 2916 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $6$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.3.0.1, 3.4.2.1, 3.6.0.1, 3.6.3.1, 3.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.6.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{6} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{2} + \left(-3 t^{3} - 3 t + 3\right) x - 3 t^{4} - 3 t^{3} + 3 t^{2} + 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $6$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$ |