Defining polynomial
|
\(x^{6} + 38\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19})$ |
| Root number: | $i$ |
| $\card{ \Gal(K/\Q_{ 19 }) }$: | $6$ |
| This field is Galois and abelian over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{19})$, 19.3.2.3 |
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: |
\( x^{6} + 38 \)
|
Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + z^{2} + 15z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |