j-invariant

Remarks

The cusp form $\Delta$ has no zeros on $\mathbb{H}$, so $j$ is holomorphic on $\mathbb{H}$. Both $E_4^3$ and $\Delta$ have weight $12$, so $j(\gamma \tau) = j(\tau)$ for every $\gamma$ in the modular group. With $q = e^{2 \pi i \tau}$, $j(\tau) = q^{-1} + 744 + 196884 q + \cdots$, so $j$ is a meromorphic modular function with a simple pole at infinity. Example: $E_6(i) = 0$, so $j(i) = 1728$; and $E_4(e^{2 \pi i/3}) = 0$, so $j(e^{2 \pi i /3}) = 0$. Non-example: $j$ is not a modular form of weight $0$, because it is unbounded as $\operatorname{Im} \tau \to \infty$. For the elliptic curve $y^2 = x^3 + ax + b$, the $j$-invariant is $1728 \cdot 4a^3 / (4a^3 + 27b^2)$.