Defining polynomial
|
\(x^{12} + 6 x^{11} + 3 x^{10} + 9 x + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{9}{4}]$ |
| Visible Swan slopes: | $[\frac{5}{4}]$ |
| Means: | $\langle\frac{5}{6}\rangle$ |
| Rams: | $(5)$ |
| Jump set: | $[2, 14]$ |
| Roots of unity: | $6 = (3 - 1) \cdot 3$ |
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$, 3.1.4.3a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 6 x^{11} + 3 x^{10} + 9 x + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^9 + z^6 + 1$,$z^2 + 2$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
| Galois degree: | $216$ |
| Galois group: | $S_3^2:C_6$ (as 12T121) |
| Inertia group: | $C_3^2:C_{12}$ (as 12T73) |
| Wild inertia group: | $C_3^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[2, \frac{9}{4}, \frac{9}{4}]$ |
| Galois Swan slopes: | $[1,\frac{5}{4},\frac{5}{4}]$ |
| Galois mean slope: | $2.175925925925926$ |
| Galois splitting model: | $x^{12} - 2 x^{9} + 3 x^{6} + 4 x^{3} + 1$ |