Plane quartics over $\mathbb {Q}$ with complex multiplication
2018
We give examples of smooth plane quartics over QQQ with complex multiplication over Q¯¯¯¯Q¯\overline{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.
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