  We prove asymptotic formulas for cyclicity of reductions of elliptic curves
over the rationals in a family of curves having specified torsion. These
results agree with established conditional results and with average results
taken over larger families. As a key tool, we prove an analogue of a result of
Vl\u{a}du\c{t} that estimates the number of elliptic curves over a finite field
with a specified torsion point and cyclic group structure.
