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For an algebraic K3 surface with complex multiplication (CM), algebraicfibres of the associated twistor space away from the equator are again of CMtype. In this paper, we show that algebraic fibres corresponding to points atthe same altitude of the twistor base $S^2\simeq \mathbb{P}^1_\mathbb{C}$ sharethe same CM endomorphism field. Moreover, we determine all the admissiblePicard numbers of the twistor fibres.

Complex Multiplication and Noether-Lefschetz Loci of the Twistor Space of a K3 Surface