Ramification of the derivative of the Weierstrass elliptic function ($\mathrm{deg}{\wp }^{\prime }=3,{\sum }_p(e_p-1)=6$)

On its period torus, the derivative has one triple pole, so degree three. The Riemann-Hurwitz formula gives total ramification six, counting each ramification point with weight $e_p-1$. The number of distinct ramification points is five when $g_2\ne 0$ and three when $g_2=0$. Indeed ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$ by the Weierstrass elliptic differential equation. If $g_2\ne 0$, each of its two possible $\wp$-values has two distinct unramified preimages for $\wp$, because coincidence with a cubic branch root would force a repeated root. These four simple zeros of ${\wp }^{\prime \prime }$ give local degree two for ${\wp }^{\prime }$, and its triple pole contributes local degree three. If $g_2=0$, nonsingularity forces $g_3\ne 0$; the two simple zeros of $\wp$ are double zeros of ${\wp }^{\prime \prime }$ and each gives local degree three for ${\wp }^{\prime }$. Along with its pole these are three distinct points, each of weight two. A hexagonal period lattice realizes this second case by its sixth-root rotational symmetry.