Defining polynomial
| \( x^{15} + 15 x^{14} + 5 x^{13} + 20 x^{12} + 15 x^{11} + 12 x^{10} + 20 x^{9} + 15 x^{8} + 5 x^{7} + 20 x^{6} + 12 x^{5} + 15 x^{4} + 15 x^{2} + 20 x + 2 \) |
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(5 t + 10\right) x^{4} + \left(20 t^{2} + 15 t + 5\right) x^{3} + \left(5 t^{2} + 10 t + 5\right) x^{2} + \left(5 t^{2} + 5 t + 5\right) x + 20 t^{2} + 20 t + 10 \in\Q_{5}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_5^2:C_{12}$ (as 15T19) |
| Inertia group: | Intransitive group isomorphic to $C_5^2:C_4$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | [5/4, 5/4] |
| Galois mean slope: | $123/100$ |
| Galois splitting model: | $x^{15} - 20 x^{13} - 60 x^{12} - 10 x^{11} + 2556 x^{10} - 750 x^{9} - 15260 x^{8} - 11235 x^{7} + 21760 x^{6} + 69046 x^{5} - 233900 x^{4} + 575 x^{3} + 1239400 x^{2} - 129840 x + 488384$ |