Does India have its own rocket?

Yes, India does have its own rockets. They are PSLV and GSLV. PSLV is considered to be the workhorse of ISRO with a very high success rate. You can read about the GSLV Mk III – A Comprehensive Overview of Heavy Lift Launch Vehicle in the given link.

Further readings:

1. Mars Orbiter Mission (MOM) – Payloads, Launch Vehicle, Achievements
2. Reusable Launch Vehicle – Technology Demonstrator (RLV – TD)