= Solution
For a nonconstant <holomorphic map> of degree $d$ between compact connected <Riemann surfaces>,
$$
\boxed{2g_X-2=d(2g_Y-2)+\sum_{p\in X}(e_p-1),}
$$
where the <analytic ramification index> $e_p$ is the exponent in local coordinates $w\circ f=z^{e_p}$ after coordinate changes. A <meromorphic function> maps its <poles> to infinity; the target coordinate $1/w$ is then holomorphic and vanishes at a <pole>, so this defines a <holomorphic map> to the <Riemann sphere>. A <pole> of order $m$ has local degree $m$.
Let $\Lambda$ be the <period lattice>. The induced function $\wp:T=\mathbb C/\Lambda\to\mathbb{CP}^1$ has just one <pole>, of order two, so its degree is two. The <Riemann-Hurwitz formula> gives total ramification four. The <pole> contributes one; every <analytic ramification index> is at most two, so there are exactly three further <ramification points>, each simple. Evenness of $\wp$, obtained by reindexing its lattice sum, identifies them with the three nonzero half-periods. Their values $e_1,e_2,e_3$ are distinct: two points of local degree two cannot lie in one degree-two fiber.
The functions $\wp'^2$ and $\prod_i(\wp-e_i)$ have double zeros at these same three points and a <pole> of order six at zero. Their ratio is a holomorphic nowhere-zero function on the compact <torus>, hence constant. From $\wp(z)=z^{-2}+O(z^2)$ and $\wp'(z)=-2z^{-3}+O(z)$ its value is four. Therefore
$$
\boxed{\wp'^2=4(\wp-e_1)(\wp-e_2)(\wp-e_3),\qquad e_i\ne e_j\ (i\ne j).}
$$
\b[The printed additional assertion that all $e_i$ are nonzero is false.] The <square-lattice Weierstrass function has a zero branch value>: for $\mathbb Z+i\mathbb Z$, reindexing gives $\wp(iz)=-\wp(z)$. At the half-period $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp(h)=-\wp(h)$ and hence $\wp(h)=0$. Thus one branch value is zero, while the three remain distinct.
The <derivative> has a single triple <pole>, so \b[$\deg\wp'=3$], and the <Riemann-Hurwitz formula> gives \b[total ramification $\sum(e_p-1)=6$]. The number of distinct <ramification points> requires a further distinction. The absent constant term in the <Laurent expansion> gives $e_1+e_2+e_3=0$, so write $\wp'^2=4\wp^3-g_2\wp-g_3$. Differentiation yields $\wp''=6\wp^2-g_2/2$. If $g_2\ne0$, each of the two values $\pm\sqrt{g_2/12}$ has two distinct preimages under $\wp$ and none is a branch value, since a common zero of the cubic and its <derivative> would be a repeated cubic root. Thus there are four simple finite <ramification points> of $\wp'$ and its triple <pole>, giving \b[five distinct points]. If $g_2=0$, the nonsingularity of the cubic forces $g_3\ne0$. The two simple zeros of $\wp$ make $\wp''$ vanish to order two, so each has <analytic ramification index> three for $\wp'$. Together with its triple <pole> there are \b[three distinct points]. This second case occurs for a hexagonal lattice: its sixth-root rotational symmetry forces the $z^2$ Laurent coefficient, and hence $g_2$, to vanish. The weighted count is six in both cases.
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