Defining polynomial
|
\(x^{6} + 166\)
|
Invariants
| Base field: | $\Q_{83}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{83}(\sqrt{83})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 83 }) }$: | $2$ |
| This field is not Galois over $\Q_{83}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{83}(\sqrt{83})$, 83.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{83}$ |
| Relative Eisenstein polynomial: |
\( x^{6} + 166 \)
|
Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 6T3) |
| Inertia group: | $C_6$ (as 6T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: |
$x^{6} - 18 x^{5} + 135 x^{4} - 540 x^{3} + 1215 x^{2} - 1458 x + 646$
|