Defining polynomial
| \( x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $15$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $17$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $1$ |
| This field is not Galois over $\Q_{5}$. | |
Intermediate fields
| 5.3.2.1, 5.5.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: | \( x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10 \) |
Invariants of the Galois closure
| Galois group: | $S_3\times F_5$ (as 15T11) |
| Inertia group: | $F_5\times C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | [5/4] |
| Galois mean slope: | $71/60$ |
| Galois splitting model: | $x^{15} - 5 x^{14} + 20 x^{13} - 65 x^{12} + 165 x^{11} - 338 x^{10} + 610 x^{9} - 1315 x^{8} + 4780 x^{7} - 8080 x^{6} + 7248 x^{5} + 4645 x^{4} - 12205 x^{3} + 2035 x^{2} + 8290 x + 2089$ |