Zeta function of an elliptic curve over a finite field
= Zeta function of an elliptic curve over a finite field
{title2=$Z_E(T)$}
This finite-field function supplies a local factor of a <Hasse–Weil zeta function> when the curve is a reduction of a curve over a number field.
If $a=q+1-\#E(\mathbb F_q)$, then
$$
Z_E(T)=\exp\left(\sum_{r\geq1}\#E(\mathbb F_{q^r})\frac{T^r}{r}\right)
=\frac{1-aT+qT^2}{(1-T)(1-qT)}.
$$