Sheldon Katz has computed this in general in On the finiteness of rational curves on quintic threefolds. See Appendix B on pp. 158-159. He gives a list of the possibilities based on the equations. It is rather simple to check the condition. According to my computation the answer in the above case is $\mathcal O_C(-1)\oplus O_{\mathbb P^1}(-1)\oplus \mathcal O_C(-1)$O_{\mathbb P^1}(-1)$, but it is easy to produce explicit examples with other normal bundles. 2 added 78 characters in body; added 103 characters in body Sheldon Katz has computed this in general in On the finiteness of rational curves on quintic threefolds. See Appendix B on pp. 158-159. You need to do He gives a little work, but given explicit list of the possibilities based on the equations, it's a tedious. It is rather simple to check the condition. According to my computation the answer in the above case is$\mathcal O_C(-1)\oplus \mathcal O_C(-1)\$, but it is easy computationto produce explicit examples with other normal bundles.