Let $S\overset{\pi}{\to} E$ be a ruled surface over an elliptic curve over complex field. Clearly, there are rational curves and elliptic curves on $S$. Is there any higher genus curves on $S$. Are all the elliptic curves isomorphic. The reason for the second question is that, all the smooth sections are isomorphic elliptic curves, except this sections is there any other elliptic curves?If so, how do I find them? 



Curves on elliptic ruled surfaces?
Any surface has lots of curves of high genus. Just take a generic hypersurface section of high degree.


Any other elliptic curve will be isogenous to $E$ with $\pi$ inducing the isogeny. I think you can embed any isogenous curve in $S$.


