Defining polynomial
|
\(x^{6} + 3 x + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/4]$ |
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.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{6} + 3 x + 3 \)
|
Ramification polygon
| Residual polynomials: | $2z + 2$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $\SOPlus(4,2)$ (as 6T13) |
| Inertia group: | $C_3^2:C_4$ (as 6T10) |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4, 5/4]$ |
| Galois mean slope: | $43/36$ |
| Galois splitting model: |
$x^{6} + 3 x + 3$
|