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We work towards completely classifying all bielliptic Shimura curves$X_0^D(N)$ with nontrivial level $N$, extending a result of Rotger thatprovided such a classification for level one. Combined with prior work, thisallows us to determine the list of all pairs $(D,N)$ for which $X_0^D(N)$ hasinfinitely many degree $2$ points, up to $2$ explicit possible exceptions. Asan application, we use these results to make progress on determining whichcurves $X_0^D(N)$ have sporadic points.

Bielliptic Shimura curves $X_0^D(N)$ with nontrivial level