On the Curve X(9)

Y. Kopeliovich, J. Quine

In this note we use the machinery of computing with theta functions to describe a model for the modular curve $H/\Gamma(9)}$, also known as $X(9)$. We show that this curve of genus 10 is given as a complete intersection of two cubics. We construct differentials on the curve through the use of theta functions with characteristics. Finally we use this information to find the Weierstrass gap sequence for the Weierstrass points.