Defining polynomial
|
\(x^{15} + 38\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 19 }) }$: | $3$ |
| This field is not Galois over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| 19.3.2.3, 19.5.4.1 |
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: |
\( x^{15} + 38 \)
|
Ramification polygon
| Residual polynomials: | $z^{14} + 15z^{13} + 10z^{12} + 18z^{11} + 16z^{10} + z^{9} + 8z^{8} + 13z^{7} + 13z^{6} + 8z^{5} + z^{4} + 16z^{3} + 18z^{2} + 10z + 15$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_5$ (as 15T3) |
| Inertia group: | $C_{15}$ (as 15T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $15$ |
| Wild slopes: | None |
| Galois mean slope: | $14/15$ |
| Galois splitting model: | Not computed |