Class number in constant extensions of elliptic function fields

For $F/K$ a function field of genus one having the finite field $K$ as field of constants and $E$ the constant extension of degree $n$ we give explicitly the class number of the field $E$ as a polynomial expression in terms of the class number of $F$ and the order of the field $K$. Applications are made to determine the degree of a constant extension $E$ necessary to have a predetermined prime $p$ occur as a divisor of the class number of the field $E$.