It is often claimed that the following equation holds for the modular discriminant $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the Dedekind eta ($\eta$) function for period ratio $\tau=\omega_2/\omega_1$:
$\Delta(\tau)=(2\pi)^{12}\eta^{24}(\tau)$
But this can't apply to all values of the invariants $g_2$ and $g_3$ because for the scaled elliptic function with periods $\lambda\omega_1$ and $\lambda\omega_2$ the period ratio and thus $\eta(\tau)$ are the same but the invariants $g_2(\lambda\omega_1,\lambda\omega_2)=\lambda^{-4}g_2(\omega_1,\omega_2)$ and $g_3(\lambda\omega_1,\lambda\omega_2)=\lambda^{-6}g_2(\omega_1,\omega_2)$ and the modular discriminant $\Delta(\lambda\omega_1,\lambda\omega_2)=\lambda^{-12}g_2^3-27\lambda^{-12}g_3^2$ are different. For a given period ratio, for which values of $\omega_1$ and $\omega_2$ do the equation apply? Is it when the periods are the half-periods of the Jacobi elliptic functions (also known as twice the Legendre complete elliptic integrals of the first kind)? Or is it when $\omega_1=1$? Wolfram Functions says that it's the latter, but I'm not sure if it's an authoritative enough source. Neither the Wikipedia articles linked above nor DLMF make it clear.