Plane quartics over $\mathbb {Q}$ with complex multiplication
Volume 185 / 2018
Acta Arithmetica 185 (2018), 127-156
MSC: 13A50, 14H25, 14H45, 14K22, 14Q05, 11G15, 14K25, 11Y40.
DOI: 10.4064/aa170227-16-3
Published online: 13 July 2018
Abstract
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.
