On congruent zeta of elliptic curves

Cosider a curve $E:y^2=x(x-1)(x-\lambda) \qquad( \lambda \in \mathbb{F}_q)$ . then:

$$\char93 E(\mathbb{F}_{q^r}) = \sum_{x\in \mathbb{F}_{q^r}} ((x(x-1)(x-\lambda))^ {\frac{q-1}{2}} +1) +1$$

$$= q+1+\sum_{x\in \mathbb{F}_{q^r}} x^{\frac{q-1}{2}} ((x-1)(x-\lambda ^{\frac{q-1}{2}}$$

We have on the other hand:

LEMMA 6.6  

$$% latex2html id marker 817\sum_{x \in \mathbb{F}_{q^r}} x^... ...)\vert k$ and $k\neq 0$.} \\ 0 & \text{otherwise.} \end{cases}$$

$$\char93 E(\mathbb{F}_{q^r}) = q+1- \o... ...orname{coeff}\left( (x-1)(x-\lambda)\right)^{\frac{q-1}{2}}, x^{\frac{q-1}{2}})$$

Further computations are found in Clemens: ``A scrapbook of complex curve thery''. Students that are interested in this subject are advised to read the book.