= Solution
The <Riemann-Hurwitz formula> gives total ramification four because $0=2\cdot2-R$. The double pole at $[0]$ has local degree two and contributes one. Since $\wp'$ is odd and periodic, it vanishes at each nonzero point satisfying $[h]=[-h]$. These are
$$
\boxed{\frac{\omega_1}{2},\quad\frac{\omega_2}{2},\quad
\frac{\omega_1+\omega_2}{2}\pmod\Lambda.}
$$
They are three distinct finite points, each contributing at least one to ramification. Together with $[0]$ they exhaust the total of four. Each therefore has local degree exactly two, meaning its derivative has a simple zero. There are \b[no other zeros of $\wp'$], and their translates by $\Lambda$ give the full list in the complex plane.
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