Defining polynomial
\(x^{12} - 51 x^{9} + 867 x^{6} - 4913 x^{3} + 111166451\)  |
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: | $1$ |
| $|\Gal(K/\Q_{ 17 })|$: | $12$ |
| This field is Galois over $\Q_{17}.$ | |
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 \( x^{4} - x + 11 \) |
| Relative Eisenstein polynomial: | \( x^{3} - 17 t^{3} \)$\ \in\Q_{17}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |