Student data in a database
I'm no big fan of using partial classes because I find it less readable. Like the other commenter I think you should look at the repository pattern for a cleaner solution.
It seems a bit odd that you give Student the responsibility to save itself to the context. 

Something in the same neighbourhood is the UnitOfWorkPattern. That said the DbContext is an implementation of UnitOfWork and Repository Pattern. You could use the DbContext directly or with a wrapper of your own, but if your wrapper doesn't add value to the DbContext you could skip it.

You are initializing your DbContext multiple times which seems redundant. Most likely you get the same instance since it implements the UnitOfWork pattern, but the "new" and private variables combined with static methods will make it very hard to write unit-tests. For testability you should avoid having MyDbContext in a private variable. You can use constructor injection to make it available to the class, or keep it in a property that you can substitute in tests later.