# Plane quartics over $\mathbb{Q}$ with complex multiplication

## TL;DR

This paper constructs explicit examples of smooth plane quartic curves over the rationals with complex multiplication, detailing algorithms for their computation and analyzing their reduction properties.

## Contribution

It provides explicit examples of plane quartics with primitive CM over $ar{bQ}$ and develops algorithms for their construction and analysis.

## Key findings

- Explicit examples of CM plane quartics over $bQ$
- Algorithms for period matrix reduction and invariant computation
- Analysis of reduction properties of the constructed curves

## 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.