Elliptic ElGamal Public Key Cryptosystem doubt
In points of elliptic curve instead of $\frac {a}{b} $ you should use $a\cdot b^{-1} \pmod p$. Also: $$\sqrt3=\{i \in [0,12]|i^2=3 \pmod {13}\}=\{4,9\}$$  So $P=(6,4)$ or $P=(6,9)$ which $2\cdot P$ are $(10,8)$ and $(10,5)$ respectively. But your computation show that $2\cdot P=(12,\sqrt {23})$.


Also with $x=12$ we have't a solution for $y$ in your elliptic curve.