Elementary matrix decomposition and the computation of Darmon points with higher conductor

We extend the algorithm of [DG02] and [DP06] for computing p-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of [DL03] for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups Γ1(N) for ideals N in certain rings of S-integers. We use these extensions to provide additional evidence in support of the conjectures on the rationality of Darmon points.