Seminal works by Birch and Ihara gave formulas for the mth power moments of the traces of Frobenius endomorphisms of elliptic curves over Fp for primes p≥5. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a finite abelian group A. We revisit these formulas and determine a simple expression for the zeta function Zp(A;t), the generating function for these mth power moments. In particular, we find thatZp(A;t)=Zˆp(A;t)∏a∈Frobp(A)(1−at), where Frobp(A):={a:−2p≤a≤2p and a≡p+1(mod|A|)}, and Zˆp(A;t) is an easily computed polynomial that is determined by the first ⌈2⌊2p⌋|A|⌉ power moments. These rational zeta functions have two natural applications. We find rational generating functions in weight aspect for traces of Hecke operators on Sk(Γ) for various congruence subgroups Γ. We also prove congruence relations for power moments by making use of known congruences for traces of Hecke operators.