Defining polynomial
| \( x^{12} + 6 x^{11} - 9 x^{10} + 12 x^{9} + 12 x^{8} - 9 x^{7} + 3 x^{6} - 9 x^{5} + 9 x^{4} + 6 x^{3} - 9 x^{2} - 9 x + 6 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $19$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $6$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.4.3.2, 3.6.9.7 |
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: | \( x^{12} + 6 x^{11} - 9 x^{10} + 12 x^{9} + 12 x^{8} - 9 x^{7} + 3 x^{6} - 9 x^{5} + 9 x^{4} + 6 x^{3} - 9 x^{2} - 9 x + 6 \) |
Invariants of the Galois closure
| Galois group: | $C_3\times C_3:D_4$ (as 12T42) |
| Inertia group: | $C_3\times (C_3 : C_4)$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | [3/2, 2] |
| Galois mean slope: | $7/4$ |
| Galois splitting model: | $x^{12} - 39 x^{10} - 14 x^{9} + 408 x^{8} - 462 x^{7} - 2913 x^{6} + 4074 x^{5} + 5898 x^{4} - 19712 x^{3} - 2949 x^{2} + 19236 x - 3821$ |