Defining polynomial
\(x^{12} - 2 x^{11} + 4 x^{10} + 4 x^{9} - 4 x^{7} + 10 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $16$ |
Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
Root number: | $i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, 2.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{6} + 2 t x^{5} + 2 x^{3} + 2 x^{2} + 2 t + 4 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t + 1$,$z^{4} + z^{2} + 1$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
Galois group: | $A_4^2:D_4$ (as 12T208) |
Inertia group: | Intransitive group isomorphic to $C_2^2\wr C_3$ |
Wild inertia group: | $C_2^6$ |
Unramified degree: | $6$ |
Tame degree: | $3$ |
Wild slopes: | $[4/3, 4/3, 4/3, 4/3, 2, 2]$ |
Galois mean slope: | $175/96$ |
Galois splitting model: | $x^{12} - 12 x^{10} - 44 x^{9} + 24 x^{8} + 276 x^{7} + 558 x^{6} + 192 x^{5} - 1860 x^{4} - 3356 x^{3} - 1776 x^{2} - 168 x - 4$ |