Defining polynomial
| \( x^{12} + 500 x^{6} - 3125 x^{2} + 62500 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $6$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 5 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{*})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5*})$, 5.3.0.1, 5.4.2.1, 5.6.0.1, 5.6.3.1, 5.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.6.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{6} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 5 t^{2} \in\Q_{5}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $6$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{12} - x^{11} + 2 x^{10} - 3 x^{9} + 5 x^{8} - 8 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1$ |