Defining polynomial
\(x^{12} + 2 x^{3} + 2 x^{2} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[4/3, 4/3]$ |
Intermediate fields
2.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{12} + 2 x^{3} + 2 x^{2} + 2 \) |
Ramification polygon
Residual polynomials: | $z^{3} + z + 1$,$z^{8} + z^{4} + 1$ |
Associated inertia: | $3$,$2$ |
Indices of inseparability: | $[3, 2, 0]$ |