Defining polynomial
|
\(x^{14} + 19\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 19 }) }$: | $2$ |
| This field is not Galois over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{19\cdot 2})$, 19.7.6.1 |
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: |
\( x^{14} + 19 \)
|
Ramification polygon
| Residual polynomials: | $z^{13} + 14z^{12} + 15z^{11} + 3z^{10} + 13z^{9} + 7z^{8} + z^{7} + 12z^{6} + z^{5} + 7z^{4} + 13z^{3} + 3z^{2} + 15z + 14$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times F_7$ (as 14T7) |
| Inertia group: | $C_{14}$ (as 14T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $14$ |
| Wild slopes: | None |
| Galois mean slope: | $13/14$ |
| Galois splitting model: | Not computed |