Defining polynomial
| \( x^{15} + 24 x^{14} + 12 x^{13} + 3 x^{12} + 24 x^{11} + 6 x^{10} + x^{9} + 18 x^{8} + 12 x^{7} + 18 x^{6} + 3 x^{5} + 12 x^{4} + 19 x^{3} + 15 x^{2} + 9 x + 7 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $5$ |
| Discriminant exponent $c$ : | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| 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} + \left(3 t^{4} + 3 t^{3} + 3 t^{2} + 3 t\right) x^{2} + \left(3 t^{4} + 3\right) x + 3 t^{3} + 6 t^{2} + 6 t + 3 \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: | $x^{15} - 3 x^{14} - 2946 x^{13} + 57615 x^{12} + 2069277 x^{11} - 77319594 x^{10} + 223873337 x^{9} + 27148718595 x^{8} - 540347561913 x^{7} + 2718657893993 x^{6} + 65089944363519 x^{5} - 1693692847776387 x^{4} + 20530684903788623 x^{3} - 144656138022354831 x^{2} + 540008298412833288 x - 732754866776488181$ |