= Solution
Multiplication by two on the <complex torus> is a holomorphic unramified covering
$$
[2]:T\to T,\qquad[z]\mapsto[2z].
$$
Its <kernel> is $(\tfrac12\Lambda)/\Lambda$, of order four; its derivative is the nonzero constant two. Therefore $\deg[2]=4$. Composition with the degree-two <Weierstrass elliptic function> gives
$$
\boxed{\deg\bigl([z]\mapsto\wp(2z)\bigr)=4\cdot2=8.}
$$
It is well-defined because changing $z$ by $\omega\in\Lambda$ changes $2z$ by the period $2\omega$.
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