Defining polynomial
| \( x^{15} + 5 x^{14} + 15 x^{12} + 15 x^{11} + 2 x^{10} + 5 x^{9} + 20 x^{8} + 2 x^{5} + 5 x^{4} + 5 x^{3} + 20 x^{2} + 20 x + 22 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| 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
| 5.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{5} + \left(15 t^{2} + 15 t + 15\right) x^{4} + \left(10 t^{2} + 5 t + 20\right) x^{3} + \left(5 t^{2} + 5 t + 5\right) x^{2} + \left(10 t^{2} + 5\right) x + 5 t^{2} + 5 t + 15 \in\Q_{5}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 15T38 |
| Inertia group: | Intransitive group isomorphic to $C_5^2:D_5.C_2$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | [5/4, 5/4, 5/4] |
| Galois mean slope: | $623/500$ |
| Galois splitting model: | $x^{15} - 195 x^{13} + 15210 x^{11} - 1586 x^{10} - 604175 x^{9} + 206180 x^{8} + 12852450 x^{7} - 9381190 x^{6} - 138705567 x^{5} + 174222100 x^{4} + 568946105 x^{3} - 1132443650 x^{2} + 447265260 x + 79953224$ |