Defining polynomial
|
\(x^{18} + 19\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $18$ |
| Ramification exponent $e$: | $18$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $17$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
| Root number: | $i$ |
| $\card{ \Gal(K/\Q_{ 19 }) }$: | $18$ |
| This field is Galois and abelian over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{19\cdot 2})$, 19.3.2.2, 19.6.5.5, 19.9.8.8 |
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: |
\( x^{18} + 19 \)
|
Ramification polygon
| Residual polynomials: | $z^{17} + 18z^{16} + z^{15} + 18z^{14} + z^{13} + 18z^{12} + z^{11} + 18z^{10} + z^{9} + 18z^{8} + z^{7} + 18z^{6} + z^{5} + 18z^{4} + z^{3} + 18z^{2} + z + 18$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{18}$ (as 18T1) |
| Inertia group: | $C_{18}$ (as 18T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $18$ |
| Wild slopes: | None |
| Galois mean slope: | $17/18$ |
| Galois splitting model: | Not computed |