I drew a spirograph curve $X = \{ (\cos \theta + 0.5 \cos 5 \theta, \sin \theta + 0.5 \sin 5 \theta): \theta \in [0,2\pi] \}\subseteq \mathbb{R}^2 $ I would like to know what is the genus? Is it birational to a circle? I think not because the inverse map involves solving a quintic. Do we need to complexify and consider the Riemann surface $X(\mathbb{C})$? I know the sphere $w^2 + z^2 - = 0$ has genus $g=0$.
This curve is the image of the unit circle under the map: $z \mapsto w= z \,(1 + 0.5\, z^4)$ and I'm asking if the image curve is birational to a circle.
Fixes
The comments say the answer might be yes...! Could you point me to a proof of that? This might not be elementary, the obvious map $z \mapsto f(z)$ doesn't work... so is there another map taking $X(\mathbb{Q}) \leftrightarrow \mathbb{P}^1$ ?
There might be an error in my question... I have drawn the image of the curve $\{ |z| = 1\} \subseteq \mathbb{C}$ by the curve $z \mapsto f(z)$ in the $w$-plane, and I'm asking for the genus of that image. Is that image curve birational to a circle? There are rational points $(x,y) = (\pm 1.5,0), (0, \pm 1.5)$ and we could find the intersection $\ell \cap X$ of the curve and a line with rational slope and I guess we could construct a map?