The form of the zeta function of an elliptic curve over a finite field

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!

• Well, there is a relatively elementary proof in Silverman. The Arithmetic of Elliptic Curves – Grigory M Aug 21 '15 at 18:31
• And of course a proof for $y^2=x^3-x$ goes back to Gauss (see e.g. Ireland, Rosen. A Classical Introduction to Modern Number Theory, ch. 8 for the proof) – Grigory M Aug 21 '15 at 18:37