Defining polynomial
| \( x^{12} - 9 x^{11} + 9 x^{9} - 9 x^{8} + 9 x^{7} - 12 x^{6} + 3 x^{3} + 9 x^{2} + 9 x - 12 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $23$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3*})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $2$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.3.5.1, 3.4.3.2, 3.6.11.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 9 x^{11} + 9 x^{9} - 9 x^{8} + 9 x^{7} - 12 x^{6} + 3 x^{3} + 9 x^{2} + 9 x - 12 \) |
Invariants of the Galois closure
| Galois group: | $C_3:D_4$ (as 12T13) |
| Inertia group: | $C_3 : C_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | [5/2] |
| Galois mean slope: | $23/12$ |
| Galois splitting model: | $x^{12} + 18 x^{6} - 3$ |