The Divergence of The Elliptical Integral of First Kind $F(\phi,k)$

Question

For what values of $k$ does the following elliptical integral of the first kind diverge?

$$F(\phi,k)=\int\limits_0^{tan\phi} \frac{dt}{\sqrt{(1-t^2)(1-k'^2t^2)}}$$ where $\phi=\pi/4$ and $k'^2=1-k^2.$ Observing the nature of poles of the elliptical integral yields $k'^2=1-k^2=1$ or simply $k=0$. However, do there exist other such values of $k$?