Defining polynomial
| \( x^{15} + 5 x^{14} + 20 x^{12} + 12 x^{10} + 5 x^{9} + 10 x^{8} + 15 x^{7} + 12 x^{5} + 20 x + 22 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| 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(10 t^{2} + 10 t + 15\right) x^{4} + \left(20 t^{2} + 5 t + 5\right) x^{3} + \left(15 t^{2} + 5 t + 15\right) x^{2} + \left(20 t + 20\right) x + 20 t^{2} + 5 \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: | $x^{15} + 10 x^{13} - 80 x^{12} - 35 x^{11} + 136 x^{10} + 1650 x^{9} + 700 x^{8} - 19790 x^{7} + 48380 x^{6} - 24652 x^{5} - 226000 x^{4} + 489505 x^{3} - 655820 x^{2} + 958560 x + 49216$ |