Defining polynomial
\(x^{15} + 6 x^{14} + 3 x^{13} + 18 x^{12} + 24 x^{11} + 18 x^{10} + 7 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{6} + 9 x^{5} + 3 x^{4} + 4 x^{3} + 15 x^{2} + 12 x + 25\)  |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
Intermediate fields
| 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{5} - x + 1 \) |
| Relative Eisenstein polynomial: | \( x^{3} + 3 t^{2} x^{2} + \left(6 t^{3} + 3 t^{2} + 6 t + 6\right) x + 3 t^{4} + 6 t^{3} + 3 t^{2} + 6 t + 6 \)$\ \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 15T44 |
| Inertia group: | Intransitive group isomorphic to $C_3^2:(C_3^3:C_2)$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 3/2, 3/2, 3/2, 3/2] |
| Galois mean slope: | $727/486$ |
| Galois splitting model: | $x^{15} - 15 x^{12} - 75 x^{9} + 1875 x^{6} - 6250 x^{3} - 3125$ |