Birth control HELP!

Ok girls I need some help. I want to go on birth control for period cramps. But the thing is, my mom doesn’t want me to for a few reasons. She doesn’t want me to think it’s ok to start having unprotected sex because I’m on the pill, and it will raise my chance of cancer. Cancer runs very very strongly in my family. Any tips? I have a doctor appointment next month and I want to talk to my doctor about it. My doctor has talked to me before about getting a shot (I’m not sure what that kind of birth control is called). She said I would receive the shot ever 3 months. Btw I’m 15