For any $n\geq 1$, let $E_n $ be the elliptic curve given by the equation $y^2 = x(x-1)(x-\zeta_{15^n})$. Here $\zeta_{m} = \exp(2\pi i /m)$ for any positive integer $m$.
There is a unique element $\tau_n$ in the complex upper half plane such that $E_n = \mathbf{C}/(\mathbf{Z} + \tau_n\mathbf{Z})$.
I need that $\mathrm{Im}(\tau_n) > \frac{1}{2}$. Can we show this?
It might not be true, but if it is it would help me a lot.