How to simplify $$\frac{\sqrt[3]{\vartheta _3\left(0,e^{-5 \pi }\right)}}{\sqrt[12]{2} \sqrt[6]{\vartheta _2\left(0,e^{-5 \pi }\right) \vartheta _4\left(0,e^{-5 \pi }\right)}}$$
This is a Ramanujan's Class Invariant $G(25)$.
Class invariant is defined as $G(n)=(2k(e^{-\pi\sqrt{n}})k'(e^{-\pi\sqrt{n}}))^{-1/12}$, Where $k(q)$ is a Elliptic Modulus
k[q_] := (EllipticTheta[2, 0, q]/EllipticTheta[3, 0, q])^2
kc[q_] := (EllipticTheta[4, 0, q]/EllipticTheta[3, 0, q])^2
G[n_] := (2 k[E^(-Pi Sqrt[n])] kc[E^(-Pi Sqrt[n])])^(-1/12)
G[25] // N
1.61803
I know that $G(25)$ is the golden ratio $$G(25)=\frac{1+\sqrt{5}}{2}$$ But "FullSimplify" doesn't work
G[25] // FullSimplify
Sorry for my English. Thanks!
ModularLambda[5 I] == Root[1 - 414728 #1 + 414744 #1^2 - 32 #1^3 + 16 #1^4 &, 1]
. I'm not sure how to get Mathematica to do this, however, as its modular functions are not aware of the special algebraic values. $\endgroup$