Defining polynomial
| \( x^{15} + 5 x^{14} - 10 x^{12} + 10 x^{11} + 10 x^{10} - 10 x^{9} - 10 x^{8} - 10 x^{7} - 10 x^{5} + 5 x^{4} - 10 x^{3} - 5 x^{2} - 10 x - 10 \) |
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: | \( x^{15} + 5 x^{14} - 10 x^{12} + 10 x^{11} + 10 x^{10} - 10 x^{9} - 10 x^{8} - 10 x^{7} - 10 x^{5} + 5 x^{4} - 10 x^{3} - 5 x^{2} - 10 x - 10 \) |
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: | $x^{15} - 5 x^{14} - 1305 x^{13} + 14425 x^{12} + 443435 x^{11} - 7738479 x^{10} - 17722590 x^{9} + 1066146055 x^{8} - 6587684480 x^{7} - 1964223865 x^{6} + 136632641122 x^{5} - 355500926340 x^{4} - 80127675915 x^{3} + 966211026480 x^{2} - 300974243220 x - 541837582767$ |