Defining polynomial
| \( x^{12} - 4864 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $11$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 19 })|$: | $6$ |
| This field is not Galois over $\Q_{19}$. | |
Intermediate fields
| $\Q_{19}(\sqrt{19})$, 19.3.2.1, 19.4.3.2, 19.6.5.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 4864 \) |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | $C_{12}$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | None |
| Galois mean slope: | $11/12$ |
| Galois splitting model: | Not computed |