Defining polynomial
| \( x^{12} + 21 x^{11} + 21 x^{10} - 39 x^{9} + 9 x^{8} - 36 x^{7} - 3 x^{6} + 18 x^{5} + 27 x^{4} - 27 x - 36 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.3.4.1, 3.4.2.2, 3.6.8.6 |
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} + 12 t x^{5} + \left(6 t + 3\right) x^{4} + \left(-6 t + 6\right) x^{3} + \left(9 t - 9\right) x^{2} + \left(-9 t - 9\right) x - 3 t - 9 \in\Q_{3}(t)[x]$ |