Defining polynomial
|
\(x^{12} - 30 x^{8} + 225 x^{4} + 1125\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.3.0.1, 5.4.3.1, 5.6.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{3} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 5 t^{2} \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + 4z^{2} + z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{12} - x^{11} - 12 x^{10} + 11 x^{9} + 54 x^{8} - 43 x^{7} - 113 x^{6} + 71 x^{5} + 110 x^{4} - 46 x^{3} - 40 x^{2} + 8 x + 1$ |