For each integer $D \geq 5$ with $D \equiv 0$ or $1 \bmod 4$, the Weierstrasscurve $W_D$ is an algebraic curve and a finite volume hyperbolic orbifold whichadmits an algebraic and isometric immersion into the moduli space of genus twoRiemann surfaces. The Weierstrass curves are the main examples of Teichm\"ullercurves in genus two. The primary goal of this paper is to determine the numberand type of orbifold points on each component of $W_D$. Our enumeration of theorbifold points, together with work of Bainbridge and McMullen, completes thedetermination of the homeomorphism type of $W_D$ and gives a formula for thegenus of its components. We use our formula to give bounds on the genus of$W_D$ and determine the Weierstrass curves of genus zero. We will also giveseveral explicit descriptions of each surface labeled by an orbifold point on$W_D$.
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