Defining polynomial
| \( x^{15} + 5 x^{14} - 10 x^{13} - 10 x^{12} + 5 x^{11} - 5 x^{10} - 10 x^{9} + 10 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{3} + 10 x^{2} - 5 x + 5 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $15$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $15$ |
| 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 |
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: | \( x^{15} + 5 x^{14} - 10 x^{13} - 10 x^{12} + 5 x^{11} - 5 x^{10} - 10 x^{9} + 10 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{3} + 10 x^{2} - 5 x + 5 \) |
Invariants of the Galois closure
| Galois group: | 15T27 |
| Inertia group: | $(C_5^2 : C_4):C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | [13/12, 13/12] |
| Galois mean slope: | $323/300$ |
| Galois splitting model: | $x^{15} + 15 x^{13} - 220 x^{12} - 2580 x^{11} + 14280 x^{10} - 16700 x^{9} - 454800 x^{8} + 326295 x^{7} + 2442440 x^{6} + 21389325 x^{5} + 62883660 x^{4} + 191239960 x^{3} + 129110160 x^{2} - 41569980 x + 565413680$ |