Weierstrass elliptic differential equation

The Weierstrass elliptic function satisfiesWriting $G_{2r}={\sum }_{\omega \in \Lambda \setminus \{0\}}{\omega }^{-2r}$, the Laurent coefficients of the Weierstrass elliptic function giveThus $g_2=60G_4$ and $g_3=140G_6$ cancel every nonremovable term in the Laurent series of ${\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3$ at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.

The three nonzero half-period classes make the odd derivative of the Weierstrass elliptic function vanish. Their values are distinct: two equal values would force at least four zeros for a function with only one double pole. Thus they are three distinct roots of $4X^3-g_{2}X-g_3$, whose monic discriminant gives the displayed factor. This lattice discriminant differs by $(2\pi {)}^{12}$ from the normalized modular discriminant for the lattice $\mathbb{Z}+\mathbb{Z}\tau$.