Defining polynomial
\(x^{12} + 39\) |
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
$\Q_{13}(\sqrt{13})$, 13.3.2.1, 13.4.3.2, 13.6.5.3 |
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: | \( x^{12} + 39 \) |
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: | $[0]$ |