Defining polynomial
|
\(x^{15} - 75 x^{12} - 75 x^{11} + 15 x^{10} + 1200 x^{9} + 3075 x^{8} + 1200 x^{7} + 37625 x^{6} + 108450 x^{5} + 108000 x^{4} + 47000 x^{3} + 7875 x^{2} - 1875 x + 125\)
|
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: | $1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[5/4]$ |
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
\( x^{3} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + \left(15 t^{2} + 5\right) x^{2} + \left(15 t^{2} + 5 t + 5\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2t^{2} + 4t + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5^2:C_{12}$ (as 15T19) |
| Inertia group: | Intransitive group isomorphic to $C_5:F_5$ |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4, 5/4]$ |
| Galois mean slope: | $123/100$ |
| Galois splitting model: |
$x^{15} - 20 x^{13} - 60 x^{12} - 10 x^{11} + 2556 x^{10} - 750 x^{9} - 15260 x^{8} - 11235 x^{7} + 21760 x^{6} + 69046 x^{5} - 233900 x^{4} + 575 x^{3} + 1239400 x^{2} - 129840 x + 488384$
|