Defining polynomial
|
\(x^{12} + 18 x^{4} - 27\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $6$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.3.0.1, 3.4.3.2, 3.6.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 3 t \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + z^{2} + 1$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{12} - 5 x^{11} + 6 x^{10} + 9 x^{9} - 26 x^{8} + 15 x^{7} - x^{6} + 12 x^{5} - 6 x^{4} - 16 x^{3} + 5 x^{2} + 6 x + 1$ |