Defining polynomial
|
\(x^{6} + 3 x^{5} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $6$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| 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: | $(\frac{5}{2})$ |
| Jump set: | $[1, 7]$ |
| Roots of unity: | $6 = (3 - 1) \cdot 3$ |
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$ |
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: |
\( x^{6} + 3 x^{5} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^{3} + 2$,$2 z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $72$ |
| Galois group: | $\SOPlus(4,2)$ (as 6T13) |
| Inertia group: | $C_3^2:C_4$ (as 6T10) |
| Wild inertia group: | $C_3^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{9}{4}, \frac{9}{4}]$ |
| Galois Swan slopes: | $[\frac{5}{4},\frac{5}{4}]$ |
| Galois mean slope: | $2.0833333333333335$ |
| Galois splitting model: | $x^{6} - 6 x^{4} - 4 x^{3} + 9 x^{2} + 12 x + 7$ |