Primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from $Q\left(\sqrt{-23}\right)$ and $Q\left(\sqrt{-31}\right)$, which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.