I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I found $(x,y)=(\pm 58, 15)$ but missed $(\pm 4, 3)$, and now I'm unsure if these are all the solutions.
Does anyone know of a solution to this problem?