Defining polynomial
| \( x^{13} - 19 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $13$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 19 })|$: | $1$ |
| This field is not Galois over $\Q_{19}.$ | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}$ |
| Relative Eisenstein polynomial: | \( x^{13} - 19 \) |
Invariants of the Galois closure
| Galois group: | $F_{13}$ (as 13T6) |
| Inertia group: | $C_{13}$ |
| Unramified degree: | $12$ |
| Tame degree: | $13$ |
| Wild slopes: | None |
| Galois mean slope: | $12/13$ |
| Galois splitting model: | $x^{13} - 52 x^{12} + 1248 x^{11} - 18304 x^{10} + 183040 x^{9} - 1317888 x^{8} + 7028736 x^{7} - 28114944 x^{6} + 84344832 x^{5} - 187432960 x^{4} + 299892736 x^{3} - 327155712 x^{2} + 218103808 x - 67108883$ |