Abstract This paper establishes an interesting connection between the family of CMC surfaces of revolution in $$\mathbb {E}_1^3$$ E 1 3 and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in $$\mathbb {E}_1^3$$ E 1 3 , only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in $$\mathbb E^3$$ E 3 and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in $$\mathbb E^3$$ E 3 are spheres and right circular cylinders.