Computing with necklaces on elliptic curves

We present computational algorithms to work with points on the modular curve associated to the normaliser of a non-split Cartan group of prime level $p$. Rather than working with explicit equations, we represent these points using the moduli interpretation of necklaces in the $p$-torsion of elliptic curves. We use our methods to investigate for which primes $\mathrm{\ell }\mathrm{\ne }p$ two rational points with complex multiplication can have equal reduction modulo $\mathrm{\ell }$.