  We give examples of smooth plane quartics over $\mathbb{Q}$ with complex
multiplication over $\overline{\mathbb{Q}}$ by a maximal order with primitive
CM type. We describe the required algorithms as we go, these involve the
reduction of period matrices, the fast computation of Dixmier-Ohno invariants,
and reconstruction from these invariants. Finally, we discuss some of the
reduction properties of the curves that we obtain.
