Defining polynomial
| \( x^{12} + 3 x^{10} + 3 x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{3} + 3 x^{2} - 3 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $13$ |
| 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: | \( x^{12} + 3 x^{10} + 3 x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{3} + 3 x^{2} - 3 \) |
Invariants of the Galois closure
| Galois group: | $C_3\times S_3\wr C_2$ (as 12T121) |
| Inertia group: | $(C_3\times C_3):C_4$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | [5/4, 5/4] |
| Galois mean slope: | $43/36$ |
| Galois splitting model: | $x^{12} - 8 x^{9} - 3 x^{8} - 36 x^{7} - 45 x^{6} + 96 x^{5} + 36 x^{4} - 80 x^{3} - 15 x^{2} + 6 x + 1$ |