Locally imprimitive points for elliptic curves

We will discuss a joint ongoing project with F. Campagna and N. Jones.
It concerns points on elliptic curves over a number field with the property that the reduction of primes is never a generator for the reduced curve. Earlier work, conducted in part at MPIM, considered the case when the index of the reduction was always divisible by a fixed prime. The current project deals with other cases. The approach considers the affine representation modulo composite n on the 3 dimensional Z/nZ-vector space generated by the n--th roots of the points.