There is a theorem that guarantees that when the j invariant of the elliptic curve E is an algebraic integer, the elliptic curve E has a complex multiplication.
But how to construct an elliptic curve corresponding to the class number greater than 1 ? Can you give some examples of using computer algebra systems to construct elliptic curves, such as Magma, PARI/GP, SageMath?
\begin{align*} j(\sqrt{-10}\>\!) &=6^3 (65 + 27\sqrt{5}\>\!)^3\\ \color{black}{j(\sqrt{-14}\>\!)} &\color{black}{=2^3\Big(323+228 \sqrt{2}+(59+43\sqrt{2}) \sqrt{\small{7+14\sqrt{2}}\,}\,\Big)^3}\\ j\Big({\small\frac{1+\sqrt{-15}}{2}}\Big)&=-3^3\Big({\small\frac{1+\sqrt{5}}{2}}\Big)^2(5+4\sqrt{5}\>\!)^3\\ j(\sqrt{-23}\>\!) &=\Big(\frac{23035}{3}+\frac{5(18279-1075 \sqrt{69}\>\!)\sqrt[3]{{\small108+12\sqrt{69}}}}{36}\\ &\qquad\qquad\qquad+\frac{5(18279+1075 \sqrt{69}\>\!) \sqrt[3]{{\small108-12\sqrt{69}}\,}}{36}\Big)^3\\ \end{align*}
Ref: https://alexjbest.github.io/talks/singular-moduli/slides_h.pdf