Defining polynomial
\(x^{12} + 8 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{2} + 8 x + 10\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $35$ |
Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 4]$ |
Intermediate fields
$\Q_{2}(\sqrt{-2\cdot 5})$, 2.3.2.1, 2.6.11.13 |
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: | \( x^{12} + 8 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{2} + 8 x + 10 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
Associated inertia: | $1$,$1$,$2$ |
Indices of inseparability: | $[24, 12, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2^4:D_{12}$ (as 12T154) |
Inertia group: | $C_2^4:C_{12}$ (as 12T105) |
Wild inertia group: | $C_2^4:C_4$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[8/3, 8/3, 3, 23/6, 23/6, 4]$ |
Galois mean slope: | $361/96$ |
Galois splitting model: | $x^{12} + 4 x^{10} - 4 x^{8} - 44 x^{6} - 62 x^{4} - 20 x^{2} - 2$ |