Defining polynomial
| \( x^{6} + 136 x^{3} + 7803 \) |
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 17 })|$: | $6$ |
| This field is Galois over $\Q_{17}.$ | |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, 17.3.2.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{17}(\sqrt{3})$ $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{3} - 17 t^{3} \in\Q_{17}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $S_3$ (as 6T2) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |