Define the '$n$-clover function' $ \phi _n $ as the inverse of the function
$$ f_n(r) := \int_0^r \frac{dx}{\sqrt{1-x^n}} $$
This is related to studying the arc length of clover curves (among other things). It can be seen that $ \phi_2(x) $ is simply the sine function whereas for $n = 3, 4$ we have elliptic functions. However, it is asserted in Singer & Langer 2013, 'Subdividing the trefoil by origami' that for higher values of $n$, being associated with hyperelliptic curves now, the $n$-clover functions are not elliptic, but we do at least have that $ (\phi_6(x))^2 $ is elliptic.
I can see that the second half of the previous sentence follows from statements 6-10 on page 2 of Singer & Langer's paper, but I don't see why it's true that the higher-order clover functions are generally not elliptic themselves. Is it just because the associated curves are hyperelliptic? Why is that sufficient? Or is there some other reason? No complete demonstrations please, I just want a hint.