Defining polynomial
|
$( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $30$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| Visible Artin slopes: | $[\frac{15}{8}, \frac{15}{8}]$ |
| Visible Swan slopes: | $[\frac{7}{8},\frac{7}{8}]$ |
| Means: | $\langle\frac{7}{12}, \frac{7}{9}\rangle$ |
| Rams: | $(\frac{7}{8}, \frac{7}{8})$ |
| Jump set: | undefined |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{9} + 6 t x^{7} + \left(6 t + 3\right) x^{6} + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + (2 t + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[7, 6, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| 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 |