Defining polynomial
|
$( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(9 x^{5} + 9 x^{4}\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 ) + 3$
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $30$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_2$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{5}{2}]$ |
| Visible Swan slopes: | $[\frac{3}{2}]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(\frac{3}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $728 = (3^{ 6 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.1.0a1.1, 3.6.1.0a1.1, 3.3.3.15a1.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.6.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + \left(9 t^{5} + 9 t^{4} + 18\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + (t^4 + t^3 + t^2 + t + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $108$ |
| Galois group: | $C_3^2:D_6$ (as 18T42) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |