I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use of Weierstrass Normal Form (I am happy to assume that char$(\mathbb{F}_q)\neq 2,3$) would be good! I am also happy if someone can suggest a reference which only deals with a certain class of elliptic curve.
Many thanks!