Defining polynomial
|
\(x^{12} + 52\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| 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
| $\Q_{13}(\sqrt{13})$, 13.3.2.3, 13.4.3.1, 13.6.5.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 52 \)
|
Ramification polygon
| Residual polynomials: | $z^{11} + 12z^{10} + z^{9} + 12z^{8} + z^{7} + 12z^{6} + z^{5} + 12z^{4} + z^{3} + 12z^{2} + z + 12$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |