Defining polynomial
\(x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 6\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $22$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $8$ |
This field is Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 3, 7/2]$ |
Intermediate fields
$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{2\cdot 5})$, 2.4.8.4, 2.4.9.6 x2, 2.4.10.8 x2 |
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: | \( x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 6 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$ |
Associated inertia: | $1$,$1$,$1$ |
Indices of inseparability: | $[15, 10, 4, 0]$ |