Defining polynomial
\(x^{12} - 1716 x^{6} - 91260\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$, 13.3.2.1, 13.4.2.2, 13.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} + 12 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{6} + 156 t + 78 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + 2z^{3} + 7z^{2} + 2z + 6$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |