Defining polynomial
\(x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, 2.3.0.1, 2.4.4.2, 2.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} + x^{4} + x^{3} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{2} + 2 x + 4 t^{4} + 4 t^{3} + 4 t^{2} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |