Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\vartheta_{01}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1-w^{2n-1})^2.$$
Then the division values of Jacobian elliptic functions (https://dlmf.nist.gov/22) can be written as rational functions of thetanulls: $$\operatorname{sn}[\tfrac{1}{3}K(k);k] = \frac{2\,\vartheta _{00}[q(k)]^2}{3\,\vartheta _{00 }[q(k)^3]^2 + \vartheta_{00}[q(k)]^2} = \frac{3\,\vartheta_{01}[q(k)^3]^2 - \vartheta_{ 01}[q(k)]^2}{3\,\vartheta_{01}[q(k)^3]^2 + \vartheta_{01}[q(k)]^2},$$ $$\operatorname{cn}[\tfrac{2}{3}K(k);k] = \frac{3\,\vartheta _{00}[q(k)^3]^2 - \vartheta _{00 }[q(k)]^2}{3\,\vartheta _{00}[q(k)^3]^2 + \vartheta _{00}[q(k)]^2} = \frac{2\,\vartheta _{ 01}[q(k)]^2}{3\,\vartheta _{01}[q(k)^3]^2 + \vartheta _{01}[q(k)]^2},$$ $$\operatorname{sn}[\tfrac{1}{5}K(k);k] = \biggl\{\frac{\sqrt{5}\,\vartheta_{01}[q(k) ^5]}{\vartheta_{01}[q(k)]} - 1\biggr\}\biggl\{\frac{5\,\vartheta_{01}[q(k)^{10}]^2}{ \vartheta_{01}[q(k)^2]^2} - 1\biggr\}^{-1},$$ $$\operatorname{sn}[\tfrac{3}{5}K(k);k] = \biggl\{\frac{\sqrt{5}\,\vartheta_{01}[q(k) ^5]}{\vartheta_{01}[q(k)]} + 1\biggr\}\biggl\{\frac{5\,\vartheta_{01}[q(k)^{10}]^2}{ \vartheta_{01}[q(k)^2]^2} - 1\biggr\}^{-1},$$ $$\operatorname{cn}[\tfrac{2}{5}K(k);k] = \biggl\{\frac{\sqrt{5}\,\vartheta_{00}[q(k) ^5]}{\vartheta_{00}[q(k)]} + 1\biggr\}\biggl\{\frac{5\,\vartheta_{01}[q(k)^{10}]^2}{ \vartheta_{01}[q(k)^2]^2} - 1\biggr\}^{-1},$$ $$\operatorname{cn}[\tfrac{4}{5}K(k);k] = \biggl\{\frac{\sqrt{5}\,\vartheta_{00}[q(k) ^5]}{\vartheta_{00}[q(k)]} - 1\biggr\}\biggl\{\frac{5\,\vartheta_{01}[q(k)^{10}]^2}{ \vartheta_{01}[q(k)^2]^2} - 1\biggr\}^{-1}$$ where $$K(k)=\int_0^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2 x}}$$ and $$q(k)=e^{-\pi\frac{K(\sqrt{1-k^2})}{K(k)}}.$$
How is this possible? It is well-known that the Jacobian functions themselves can be written as ratios of theta functions (not thetanulls), but theta functions with zero argument (thetanulls) apparently suffice to describe their division values. The above formulas are unsourced information from Wikipedia (hence I'm writing this question).
In other words, it appears that $$\operatorname{sn}\left(\frac{aK(k)}{b};k\right)$$ is a rational function of thetanulls whenever $a/b\in\mathbb{Q}$ (or when the denominator of $a/b$ is an odd prime – this could have something to do with modular equations); the same should apply for other Jacobian elliptic functions – is it true? If so, how can that be done?
