1998 Toyota Corolla Battery

I initially had a problem with my battery last year. I drove it every day and was on a bike ride, set the alarm, came back and it wouldn't start. I tried to jump it and every time I tried the alarm would go off. I had it towed and the dealer said the battery was bad and replaced the battery and now I am too nervous to set the alarm. I bought a new car and my corolla has been sitting for at least 3 weeks (if not more) and the battery was dead. We charged the battery and drove it around for about 10-15 minutes. The next morning the battery was dead again. Do I have an issue with the alarm system draining the battery or do I just need to have the car run for longer when I charge the battery?