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.

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