Defining polynomial
| \( x^{9} + 24 x^{3} + 18 x^{2} + 9 x + 6 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $9$ |
| Ramification exponent $e$ : | $9$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $1$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| 3.3.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: | \( x^{9} + 24 x^{3} + 18 x^{2} + 9 x + 6 \) |
Invariants of the Galois closure
| Galois group: | $He_3:C_2$ (as 9T11) |
| Inertia group: | $C_3^2 : C_6$ |
| Unramified degree: | $1$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 2, 5/2] |
| Galois mean slope: | $121/54$ |
| Galois splitting model: | $x^{9} - 9 x^{7} + 27 x^{5} - 48 x^{3} + 63 x - 38$ |