Let $E(\mathbb{F}_{11})$ the elliptic curve over $\mathbb{F}_{11}$ given by the equation $y^2=x^3+2x+2$ and $P=(1,4)$. I have to compute $P+P$.
My idea: Let $f(x,y)=x^3+2x+2-y^2$
The tangent line to $E(\mathbb{F}_{11})$ in $P$ has equation $\frac{\partial f}{\partial x}(P)\cdot (x-1)+ \frac{\partial f}{\partial y}(P)\cdot (y-4)=0\Rightarrow 5\cdot(x-1)+3\cdot(y-4)=0\Rightarrow y=2\cdot(x-1)+4\Rightarrow y=2\cdot x+2 $
Now intersect the tangent line with the curve and obtain the equation: $4\cdot x^2+8\cdot x+4=x^3+2\cdot x+2\Rightarrow x^3+7\cdot x^2 +5\cdot x+9=0$.
By Viete 's relations: $x_1+x_2+x_3=4\Rightarrow 1+1+x_3=4\Rightarrow x_3=2\Rightarrow y_3=6\Rightarrow $ The points of coordinates (2,6)$\Rightarrow P+P=(2,5).$