Hypergeometric Function and Elliptic Integral

In the wiki page, complete elliptic integrals of first and second kind are related to the Hypergeometric function via: $$K(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})\\ E(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},-\frac{1}{2};1;k^{2})$$

But when evaluate the following command in Mathematica (ver.12):

Pi/2 Hypergeometric2F1[1/2, 1/2, 1, k^2]


I get: EllipticK[k^2]

The same thing happens for $$\mathstrut_{2}F_{1}(\frac{1}{2},-\frac{1}{2};1;k^{2})$$ as well. I get EllipticE[k^2] instead of EllipticE[k].

What am I missing ?