Defining polynomial
| \( x^{12} - 12 x^{10} - 9 x^{8} - 9 x^{7} - 6 x^{6} - 9 x^{5} - 6 x^{3} - 12 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $21$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $3$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.4.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 12 x^{10} - 9 x^{8} - 9 x^{7} - 6 x^{6} - 9 x^{5} - 6 x^{3} - 12 \) |
Invariants of the Galois closure
| Galois group: | $C_3\times S_3\wr C_2$ (as 12T121) |
| Inertia group: | 12T73 |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | [2, 9/4, 9/4] |
| Galois mean slope: | $235/108$ |
| Galois splitting model: | $x^{12} - 12 x^{10} - 12 x^{9} + 54 x^{8} + 108 x^{7} - 66 x^{6} - 324 x^{5} - 171 x^{4} + 384 x^{3} + 378 x^{2} - 180 x - 39$ |