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

Average Cyclicity for Elliptic Curves in Torsion Families