Defining polynomial
| \( x^{9} + 228 x^{6} + 16967 x^{3} + 438976 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$ : | $9$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 19 })|$: | $9$ |
| This field is Galois and abelian over $\Q_{19}$. | |
Intermediate fields
| 19.3.2.2, 19.3.2.1, 19.3.2.3, 19.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} - x + 4 \) |
| Relative Eisenstein polynomial: | $ x^{3} - 19 t^{3} \in\Q_{19}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3^2$ (as 9T2) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $3$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |