Defining polynomial
|
\(x^{12} - 138 x^{9} + 10051 x^{6} + 4027277 x^{3} + 6996025\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{23}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 23 }) }$: | $6$ |
| This field is not Galois over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{5})$, 23.4.0.1, 23.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 23.4.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of
\( x^{4} + 3 x^{2} + 19 x + 5 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 23 t^{2} \)
$\ \in\Q_{23}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3:C_{12}$ (as 12T19) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $12$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |