Constructing a meromorphic function I need help with the following problem.  
"Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at $B=(1,\sqrt{2})$ and one of the zeros at $A=(0,0)$."
If $C$ is given as $\mathbb{C}/\Lambda$, I can construct the associated Weierstrass's $\wp$ function and use Abel's theorem to construct a meromorphic function with prescribed poles and zeroes. Unfortunately, I couldn't use that in the problem above because I cannot calculate two things I would additionally need: the periods of the $\wp$ function, and the Abel-Jacobi map.

How about $x/(x-1)^2$ or $y/(x-1)^2$? If you want a simple zero at $(0,0)$, then you have to be careful. Similarly, you need to calculate the divisor carefully to make sure that you have a double pole at $(1,\sqrt{2})$ and not a pole of higher order. See this answer for a calculation of the divisors of the coordinate functions.
