Does the father automatically get full custody if the mother is incarcerated?

First off I am the father of the child. The mother and I were never married, but we were together when we had our child. I am on the original birth certificate stating that I am the legal father of our child. Since, the mother and I have separated and we just had a verbal agreement, as far as when I had her and when she had her. The mother has been recently incarcerated. My question is do I automatically have full custody in the court's eyes? Or do I have to file for full custody still? Thank you