Defining polynomial
|
\(x^{6} + 115\)
|
Invariants
| Base field: | $\Q_{23}$ |
|
| Degree $d$: | $6$ |
|
| Ramification index $e$: | $6$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $5$ |
|
| Discriminant root field: | $\Q_{23}(\sqrt{23})$ | |
| Root number: | $-i$ | |
| $\Aut(K/\Q_{23})$: | $C_2$ | |
| This field is not Galois over $\Q_{23}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $22 = (23 - 1)$ |
|
Intermediate fields
| $\Q_{23}(\sqrt{23})$, 23.1.3.2a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{23}$ |
|
| Relative Eisenstein polynomial: |
\( x^{6} + 115 \)
|
Ramification polygon
| Residual polynomials: | $z^5 + 6 z^4 + 15 z^3 + 20 z^2 + 15 z + 6$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $D_6$ (as 6T3) |
| Inertia group: | $C_6$ (as 6T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8333333333333334$ |
| Galois splitting model: | $x^{6} - 23$ |