Three points $(\mathrm{\wp }(z_1),{\mathrm{\wp }}^{\prime }(z_1))$, $(\mathrm{\wp }(z_2),{\mathrm{\wp }}^{\prime }(z_2))$, and $(\mathrm{\wp }(z_3),{\mathrm{\wp }}^{\prime }(z_3))$ lie on the same "line" if and only if 1;;$z_1+z_2+z_3\equiv 0\text{ mod }L$.

Therefore we have an addition law: $(\mathrm{\wp }(z_1),{\mathrm{\wp }}^{\prime }(z_1))+(\mathrm{\wp }(z_2),{\mathrm{\wp }}^{\prime }(z_2)):=(\mathrm{\wp }(z_1+z_2),-{\mathrm{\wp }}^{\prime }(z_1+z_2))$. To find an algebraic expression for $\mathrm{\wp }(z_1+z_2)$, we compute the determinant of:

But this is not so nice since ${\mathrm{\wp }}^{\prime }$ cannot be nicely given in terms of $\mathrm{\wp }$.
PROOF:

• number of Elliptic $a$-points is 2 to say that $\mathrm{\wp }(z_1)\ne \mathrm{\wp }(z_2)$ iff $z_1\ne \pm z_2$
• Draw a line through the points to find $z_3$
• Since this is the third $a$-point, the sum of $z_1,z_2,z_3$ must be zero.