What you are talking about is not viable in database table relationships in persisting the data to MS SQL database tables.
If CommonData is a child table to all the other parent tables, then it has to have a foreign-key column for each of the possible parent table records it is linked to.
EF expects that all primary-key ID properties be integers that are controlled the the database engine as it auto-increments the number for each table record inserted into a table. And in addition, EF auto populates the primary-key id of the parent record to the child record.
What is the purpose of this? As I see it, it is not an optimal design and it's not viable IMHO.