 Nonvanishing discriminant of a complex lattice

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 $\mathbb{Z}+\mathbb{Z}\tau$.