<s>[INST] Is the curve $ z = e^{i\theta}\left(\frac{7}{8} + \frac{1}{4} e^{6i\theta}\right) $ algebraic? Is this spirograph curve algebraic?  I an only write it in polar coordinates:
$$ z = e^{i\theta}\left(\frac{7}{8} + \frac{1}{4} e^{6i\theta}\right) $$
and here is a picture.  It is a six-sided rose-shaped curve, a hypotrochoid.

I read somewhere that all spirograph curves are algebraic, so this must be the solution of some polynomial equation $p(x,y) = 0$.  Then I could ask questions about this curve as a Riemann surface.
 [/INST] Given
$$ x=\frac{7}{8}\cos\theta+\frac{1}{4}\cos(7\theta),\qquad y=\frac{7}{8}\sin\theta+\frac{1}{4}\sin(7\theta) $$
we have $x^2+y^2=\frac{1}{64}\left(53+28\cos(6\theta)\right)$ and the given hypotrochoid is an algebraic curve since we may eliminate the $t$ variable in 
$$ \left\{\begin{array}{ccc} x& =& \frac{7}{8} T_1(t) + \frac{1}{4}T_7(t)\\x^2+y^2&=&\frac{53}{64}+\frac{7}{16}T_6(t).\end{array}\right.$$
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