Defining polynomial
|
\(x^{10} + 105 x^{9} + 4435 x^{8} + 94710 x^{7} + 1038805 x^{6} + 5025997 x^{5} + 5196440 x^{4} + 2466880 x^{3} + 2611955 x^{2} + 21422390 x + 88673277\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{23}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 23 }) }$: | $2$ |
| This field is not Galois over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{5})$, 23.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{23}(\sqrt{5})$ $\cong \Q_{23}(t)$ where $t$ is a root of
\( x^{2} + 21 x + 5 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 23 \)
$\ \in\Q_{23}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | Not computed |