Six unlikely intersection problems in search of effectivity

We investigate four properties related to an elliptic curve $E_t$ in Legendre form with parameter t: the curve $E_t$ has complex multiplication, $E_{-t}$ has complex multiplication, a point on $E_t$ with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. We solve these problems effectively and in certain cases also explicitly.