With an All India Rank (AIR) of 6 lakhs in JEE Main, securing a seat in top-tier National Institutes of Technology (NITs) or Indian Institutes of Information Technology (IIITs) is generally challenging, especially for the general category. However, as an ST category candidate with a rank of 13,000, you have a significantly better chance due to category-wise reservations.

At this rank, you might have opportunities in some of the newer NITs or certain branches in relatively older NITs. For example, in the past years, ST category candidates with similar ranks have secured admissions in branches like Civil Engineering, Chemical Engineering, Metallurgical Engineering, or even some branches of Electrical Engineering in several NITs. The specific NITs and branches you can get into will depend on the cutoffs for the current year, which are influenced by factors like the number of applicants, the difficulty level of the exam, and the number of available seats.

Additionally, you would have good chances in various Government Funded Technical Institutes (GFTIs). Many GFTIs have lower cutoff ranks compared to NITs and IIITs, and your ST category rank will be advantageous here as well. Some of these institutes offer a range of engineering branches. To get a more precise idea of the colleges and branches you might be eligible for, it's recommended to use JEE Main college predictor tools available online like Careers360 . https://engineering.careers360.com/jee-main-college-predictor?icn=jee-main-cp-lp&ici=search-section_c360-home These tools consider your rank, category, and state of eligibility to provide a list of potential colleges based on past year's data. Keep an eye on the Joint Seat Allocation Authority (JoSAA) counseling process and the specific cutoffs released for each round.