Let $0 < k < 1$ and $$K := \int_0^{\pi/2} \frac{1}{\sqrt{1 - k^2 \sin^2(\theta)}} \, \mathrm{d}\theta, \; \; K' := \int_0^{\pi/2} \frac{1}{\sqrt{1 - (1-k^2) \sin^2(\theta)}} \, \mathrm{d}\theta.$$
If we form the lattice $L = 4K \mathbb{Z} \oplus 4iK' \mathbb{Z}$ and let $$e_1 = \wp(2K), \; e_2 = \wp(2iK'), \; e_3 = \wp(2K + 2iK')$$ be the values of $\wp(z;L)$ at half-periods, then the Wikipedia article claims in the last section that $$k^2 = \frac{e_2 - e_3}{e_1 - e_3}.$$ I have seen a similar formula in Markushevich's Theory of functions of a complex variable but it defines $k$ and $K$ in a way that makes this obvious (not via an elliptic integral). I don't know how to prove this using the definitions given.