Defining polynomial
|
\(x^{13} + 23\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $13$ |
| Ramification index $e$: | $13$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{23}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{23})$: | $C_1$ |
| 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
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$. |
Canonical tower
| Unramified subfield: | $\Q_{23}$ |
| Relative Eisenstein polynomial: |
\( x^{13} + 23 \)
|
Ramification polygon
| Residual polynomials: | $z^{12} + 13 z^{11} + 9 z^{10} + 10 z^9 + 2 z^8 + 22 z^7 + 14 z^6 + 14 z^5 + 22 z^4 + 2 z^3 + 10 z^2 + 9 z + 13$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $78$ |
| Galois group: | $C_{13}:C_6$ (as 13T5) |
| Inertia group: | $C_{13}$ (as 13T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $13$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9230769230769231$ |
| Galois splitting model: | not computed |