I'm assuming this is K-6 not college. Otherwise the "one class per student" rule doesn't make sense.
A Class entity has no attributes? Like Grade( K-6), Teacher, Level (regular, learning disabled), room number, etc.
I think what you are really after is an intersection between Class and Students. A class is made up of many students. A student is registered with one class...per year. Or, depending on how you define Class, several classes (Art, Music, Drama, etc.). In any event, during their time at the school, each student will have many classes. So SchoolYear may be an attribute of the Class-Student relationship so would be in the intersection table.
That way you track all classes for each student over time.
At any rate, you are defining a relationship between two entities: Student and Class. And the answer is No, an entity without attributes doesn't make sense.