I am reading Joe Silverman's book The Arithmetic of Elliptic Curves, pg. 140, and I am trying to compute $\alpha$ and $\beta$, the complex numbers which help to evaluate the zeta function for the elliptic curve $y^2=x^3+x+1$ over $\mathbb{F}_7$. The shape should be
$$Z(V / \mathbb{F}_q;T) = \exp(\sum_{n=1}^{\infty}\#V(F_{q^n})\frac{T^n}{n})$$ $$=\frac{(1-\alpha T)(1-\beta T)}{(1-T)(1-qT)}$$
Any help would be greatly appreciated. The fact that the polynomials appearing the numerator and denominator are quadratic in $T$ has to do with the fact that elliptic curves are one dimensional objects (over $\mathbb{C}$, so have to do with dimensions of Weil cohomology groups).
The curve has 5 points defined over $\mathbb F_7$, $$ E(\mathbb F_7) = \lbrace(0,1),(0,6),(2,2),(2,5),\infty\rbrace, $$ so $$ a_7 = 7+1-5 = 3. $$ Hence the numerator of the zeta function is $$ 1 - a_pT + pT^2 = 1 - 3T + 7T^2. $$ The roots are $$ \alpha=\frac{3+\sqrt{19}i}{14}\quad\text{and}\quad \beta=\frac{3-\sqrt{19}i}{14}.$$