Defining polynomial
\(x^{12} + 6 x^{10} + 2 x^{8} + 6 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $28$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 10/3]$ |
Intermediate fields
$\Q_{2}(\sqrt{-5})$, 2.3.2.1, 2.6.8.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{12} + 6 x^{10} + 2 x^{8} + 6 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
Associated inertia: | $1$,$1$,$2$ |
Indices of inseparability: | $[17, 6, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2^4:S_4$ (as 12T145) |
Inertia group: | $C_2^4:A_4$ (as 12T87) |
Wild inertia group: | $C_2^3:D_4$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[2, 2, 8/3, 8/3, 10/3, 10/3]$ |
Galois mean slope: | $149/48$ |
Galois splitting model: | $x^{12} - 22 x^{10} + 110 x^{8} + 484 x^{6} - 4840 x^{4} + 10648 x^{2} - 5324$ |