Defining polynomial
| \( x^{12} - 38 x^{8} + 361 x^{4} - 109744 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 19 })|$: | $6$ |
| This field is not Galois over $\Q_{19}.$ | |
Intermediate fields
| $\Q_{19}(\sqrt{19})$, 19.3.0.1, 19.4.3.2, 19.6.3.1 |
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^{4} - 19 t^{2} \in\Q_{19}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | Not computed |