Defining polynomial
|
\(x^{12} + 225 x^{10} + 98 x^{9} + 17904 x^{8} + 10824 x^{7} + 611647 x^{6} + 498390 x^{5} + 8494833 x^{4} + 11764900 x^{3} + 38205036 x^{2} + 73669974 x + 36476587\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $12$ |
| This field is Galois over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, 17.3.2.1 x3, 17.4.0.1, 17.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.4.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{4} + 7 x^{2} + 10 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 51 x + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |