Defining polynomial
|
\(x^{12} + 8 x^{10} + 72 x^{9} + 93 x^{8} + 432 x^{7} + 1608 x^{6} - 19008 x^{5} + 10115 x^{4} + 20592 x^{3} + 219376 x^{2} + 154296 x + 181359\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{23}(\sqrt{23})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 23 }) }$: | $6$ |
| This field is not Galois over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{23\cdot 5})$, 23.3.0.1, 23.4.3.1, 23.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 23.3.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 18 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 23 \)
$\ \in\Q_{23}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | Not computed |