Defining polynomial
\(x^{15} - 5 x^{14} - 5 x^{13} + 5 x^{12} - 10 x^{11} + 5 x^{10} - 5 x^{9} - 5 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} - 10 x^{4} + 10 x^{3} + 10 x^{2} - 10 x + 5\)  |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| 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: | \( x^{15} - 5 x^{14} - 5 x^{13} + 5 x^{12} - 10 x^{11} + 5 x^{10} - 5 x^{9} - 5 x^{8} + 5 x^{7} + 5 x^{6} - 5 x^{5} - 10 x^{4} + 10 x^{3} + 10 x^{2} - 10 x + 5 \) |
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: | $x^{15} - 5 x^{14} - 1140 x^{13} + 3110 x^{12} + 470165 x^{11} + 101903 x^{10} - 84431845 x^{9} - 308853455 x^{8} + 5617820830 x^{7} + 41490873310 x^{6} + 19586232533 x^{5} - 440565472815 x^{4} - 628374042290 x^{3} + 1126034023020 x^{2} + 790212143220 x + 48157299081$ |