Rare-earth hydrides can exhibit high-temperature superconductivity under high pressure. Here, we apply a crystal structure prediction method to the current record-holding $T_c$ material, LaH$_{10}$, and a candidate for even higher $T_c$, YH$_{10}$. We find a pressure-induced phase transition from the experimentally observed cubic LaH$_{10}$ phase to a new hexagonal phase at around 420 GPa. This hexagonal phase could explain experimental observations of hcp impurities in fcc samples. We find that YH$_{10}$ forms similar structures to LaH$_{10}$ and discuss the sensitivity of superconductivity calculations to the computational parameters used.
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