Given a lattice $\Lambda \subset \mathbb{C}$, we may define the corresponding Weierstrass $\wp$-function to be the unique even $\Lambda$-periodic elliptic function $\wp$ with poles only on $\Lambda$ and for which $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$. For $w \notin \Lambda$, we set

$$f(z)=\operatorname{det}\left(\begin{array}{ccc} 1 & 1 & 1 \\ \wp(z) & \wp(w) & \wp(-z-w) \\ \wp^{\prime}(z) & \wp^{\prime}(w) & \wp^{\prime}(-z-w) \end{array}\right)$$

an elliptic function with periods $\Lambda$. By considering the poles of $f$, show that $f$ has valency at most 4 (i.e. is at most 4 to 1 on a period parallelogram).

If $w \notin \frac{1}{3} \Lambda$, show that $f$ has at least six distinct zeros. If $w \in \frac{1}{3} \Lambda$, show that $f$ has at least four distinct zeros, at least one of which is a multiple zero. Deduce that the meromorphic function $f$ is identically zero.

If $z_{1}, z_{2}, z_{3}$ are distinct non-lattice points in a period parallelogram such that $z_{1}+z_{2}+z_{3} \in \Lambda$, what can be said about the points $\left(\wp\left(z_{i}\right), \wp^{\prime}\left(z_{i}\right)\right) \in \mathbb{C}^{2}(i=1,2,3) ?$