= Ramification of the derivative of the Weierstrass elliptic function
{title2=$\deg\wp'=3,\quad\sum_p(e_p-1)=6$}
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 $g_2\ne0$ and three when $g_2=0$. Indeed $\wp''=6\wp^2-g_2/2$ by the <Weierstrass elliptic differential equation>. If $g_2\ne0$, 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 $\wp''$ give local degree two for $\wp'$, and its triple <pole> contributes local degree three. If $g_2=0$, nonsingularity forces $g_3\ne0$; the two simple zeros of $\wp$ are double zeros of $\wp''$ and each gives local degree three for $\wp'$. 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.
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