Defining polynomial
| \( x^{12} - 5 x^{6} + 50 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{5}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $6$ |
| This field is not Galois over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{*})$, 5.4.2.2, 5.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}(\sqrt{*})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 5 t \in\Q_{5}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times C_3:C_4$ (as 12T19) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Unramified degree: | $6$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | $x^{12} - 480 x^{10} - 1260 x^{9} + 80100 x^{8} + 413700 x^{7} - 5076750 x^{6} - 40519500 x^{5} + 59710125 x^{4} + 1214507000 x^{3} + 2584503750 x^{2} - 3523957500 x - 12020550625$ |