  We study the BPS index for the four-dimensional rank-one N=2 superconformal
field theories H_0, H_1, H_2, E_6, E_7, E_8. We consider compactifications of
the E-string theory on T^2 in which these theories arise as low energy limits.
Using this realization we clarify the general structure of the BPS index. The
index is characterized by two exponents and a sequence of invariants. We
determine the exponents and the first few invariants.
