Weierstrass Point of a Riemann surface I have that $X$ is a compact Riemann surface defined by the curve $y^{2}=1-x^{6}$ and a point $P=(0,1) \in X$ in the usual coordinates $(x,y)$. Ultimately, I want to solve a Mittag-Leffler problem on this $X$, namely to construct a particular meromorphic function with prescribed principal parts. First, I want to find the least positive $n$ such that $h^{0}(nP)>1$ (where $h^{0}$ gives the dimension of the zeroth cohomology group). Then, using this value of $n$, I want to find an explicit rational function $f(x,y)$ such that $f \in H^{0}(X, \mathcal{O}_{D})$ where $D=nP$.
I think I could solve this first by showing $P=(0,1)$ is a Weierstrass point on $X$. That is, $P$ is Weierstrass if $h^{0}(gP)=1$ or in our case $h^{0}(2P)=1$. Equivalently, we can apply Riemann-Roch for the condition $h^{0}(K-2P)=0$, i.e. no holomorphic 1-form vanishes to order to at $P$. If $P$ is Weierstrass, then we can easily get our value of $n$. However, is $P$ Weierstrass? I'm having trouble showing this is the case. Actually, I'm a little unsure of what I have to show. 
Then, once we have our $n$, how do we actually construct our desired $f(x,y)$? Any help would be much appreciated.   
 A: The following is just to expand on my comment, and give you a hint.
Your (genus $2$) curve $X$ is hyperelliptic, so the filtration $$\mathbb C^2=H^0(X,K_X)\supsetneq H^0(X,K_X-P)\supseteq H^0(X,K_X-2P)$$ gives you two possibilities: either $h^0(K_X-2P)$ is $1$, or it is $0$. In the first case, $P$ is a Weierstrass point. In other words,
$$P\textrm{ is Weierstrass }\iff h^0(K_X-2P)=1\iff h^0(2P)=2.$$
Weierstrass point does not mean what you wrote. It rather means that there exists a holomorphic differential vanishing $g$ times at $P$, i.e. $h^0(K-gP)>0$, i.e. $h^0(gP)>1$ (in general).
Now, a basis for $H^0(X,K_X)$ is given by $$\omega_1=\frac{\textrm dx}{y},\omega_2=x\frac{\textrm dx}{y}.$$
I guess you can prove easily that $\omega_2$ vanishes at least twice at $P$, so that $P$ is a Weierstrass point.

Since you were asking for the least $n$ such that $h^0(nP)>1$, the following might be related (but I only know the result for genus $g\geq 3$): 
Theorem. For any Weierstrass point $P$ on a general curve of genus $g\geq 3$,
$$g=\min\,\{n\,|\,h^0(nP)>1\}.$$
