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.
 A: You have given a parametrization of your curve by a circle, and a circle has genus $0$, so it is a genus $0$ curve. Looking at random lines intersecting it, you should expect its total degree to be at least $6$ in $x,y$. To account for the high degrees it must have a lot of nodes (you can already see $6$ of them but there should be at least $10$ if the degree is $6$)
Since it is invariant by reflexion around the axis, the equation can be written in terms of $x^2$ and $y^2$.
In fact it is invariant by the action of a diedral group of order $12$ so if you knew by heart the subring of $\Bbb R[x,y]$ invariant by that you could say even more.
writing $e^{i\theta} = c+is$ where $c^2+s^2=1$ you can write $x=f(c)$ and $y=sg(c)$ where $f$ and $g$ are some polynomials of degree $7$ and $6$.
Then $x^2$ and $y^2$ are two polynomials of degree $7$ in $c^2$, so they must have an algebraic relation.
The dimension of the space of polynomials of degree at most $14$ in two variables is greater than the dimension of the space of polynomials of degree at most $98$ in one variable, so the curve's equation has degree at most $28$ in $x,y$.
A: 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 
$$ \left\{\begin{array}{ccc} x& =& \frac{7}{8} T_1(t) + \frac{1}{4}T_7(t)\\x^2+y^2&=&\frac{53}{64}+\frac{7}{16}T_6(t).\end{array}\right.$$
