So my question refers to families of elliptic curves over the $\AA^1_\CC\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not universal).

Some sources give an equation for such a family, namely $$E_1 := y^2 + xy  = x^3 - \frac{36}{j-1728} - \frac{1}{j-1728}$$

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Thanks to TomChurch's comments, I'm revising my question:

$$\text{Is there a way to see this family complex-analytically?}$$

For example, let $\hH$ be the upper half plane, and let $\hH^\circ$ denote $\hH$ punctured at the $\SL_2(\ZZ)$-orbits of $i$ and $e^{2\pi i/3}$. Let $\ZZ$ act on the product $\CC\times\hH^\circ$ by $$(m,n)\cdot(z,\tau) := (z + m\tau + n,\tau)$$ The quotient $\ZZ^2\backslash(\CC\times\hH^\circ)$ is an elliptic curve over $\hH^\circ$. Let $\SL_2(\ZZ)$ act on $\CC\times\hH^\circ$ by $$\gamma.(z,\tau) := \left(\frac{z}{c\tau + d},\frac{a\tau + b}{c\tau + d}\right)$$ This action descends to an action of $\SL_2(\ZZ)$ on $\ZZ^2\backslash(\CC\times\hH^\circ)$, but as TomChurch noted, $\gamma = -I$ sends $(z,\tau)\mapsto(-z,\tau)$, the fibers of this quotient are actually copies of $\mathbb{P}^1$ and are not elliptic curves.

Is there a similar construction that will actually yield a curve like $E_1$?