Calculating Galois groups of Quartics

Question

I want to look at the Galois group of $$x^4-2x^2-25$$ but I am having a lot of trouble calculating its order. We can easily solve this equation to get $$x=\pm\sqrt{1\pm\sqrt{26}}$$ So the splitting field $$L=\mathbb{Q}\left(\sqrt{1+\sqrt{26}},i\sqrt{\sqrt{26}-1}\right)$$ and using the fact that $$\frac{1}{\sqrt{1+\sqrt{26}}}=(1/5)\sqrt{\sqrt{26}-1}$$ Which suggests by the tower law that the degree of $L$ over $\mathbb{Q}$ is $8$. But I plugged this into the online Galois Group calculator http://magma.maths.usyd.edu.au/calc/ and it gave me an extension of degree $24$ ! What's gone wrong here?