Defining polynomial
\(x^{12} + 6 x^{10} + 4 x^{9} + 4 x^{8} + 6 x^{6} + 12 x^{2} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $29$ |
Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 7/2]$ |
Intermediate fields
2.3.2.1, 2.6.8.2 |
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: | \( x^{12} + 6 x^{10} + 4 x^{9} + 4 x^{8} + 6 x^{6} + 12 x^{2} + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
Associated inertia: | $1$,$1$,$2$ |
Indices of inseparability: | $[18, 6, 0]$ |
Invariants of the Galois closure
Galois group: | $C_4^3:D_6$ (as 12T185) |
Inertia group: | $C_4^3:C_6$ (as 12T141) |
Wild inertia group: | $C_4^3:C_2$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[4/3, 4/3, 2, 3, 19/6, 19/6, 7/2]$ |
Galois mean slope: | $619/192$ |
Galois splitting model: | $x^{12} + 54 x^{10} + 1074 x^{8} + 9360 x^{6} + 28980 x^{4} - 45000 x^{2} - 375000$ |