Defining polynomial
|
\(x^{22} + 506\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $22$ |
| Ramification exponent $e$: | $22$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{23}(\sqrt{23})$ |
| Root number: | $i$ |
| $\card{ \Gal(K/\Q_{ 23 }) }$: | $22$ |
| This field is Galois and abelian over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{23})$, 23.11.10.10 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{23}$ |
| Relative Eisenstein polynomial: |
\( x^{22} + 506 \)
|
Ramification polygon
| Residual polynomials: | $z^{21} + 22z^{20} + z^{19} + 22z^{18} + z^{17} + 22z^{16} + z^{15} + 22z^{14} + z^{13} + 22z^{12} + z^{11} + 22z^{10} + z^{9} + 22z^{8} + z^{7} + 22z^{6} + z^{5} + 22z^{4} + z^{3} + 22z^{2} + z + 22$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{22}$ (as 22T1) |
| Inertia group: | $C_{22}$ (as 22T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $22$ |
| Wild slopes: | None |
| Galois mean slope: | $21/22$ |
| Galois splitting model: | Not computed |