Defining polynomial
|
\(x^{12} + 8 x^{10} + 44 x^{9} + 63 x^{8} + 264 x^{7} + 550 x^{6} - 6336 x^{5} + 3843 x^{4} + 4532 x^{3} + 46454 x^{2} + 30668 x + 30982\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{13})$, 13.3.0.1, 13.4.3.2, 13.6.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.3.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 11 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 13 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |