So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ 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 $$E_1 := y^2 + xy = x^3 - \frac{36}{j-1728}x - \frac{1}{j-1728}$$

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

Does $E_1$ admit a nontrivial section? (other than the identity section)

Is there a way to see this family complex-analytically using quotients of the upper half plane?

What I mean is this: Let $\mathbb{H}$ be the upper half plane, and let $\mathbb{H}^\circ$ denote $\mathbb{H}$ punctured at the $SL_2(\mathbb{Z})$-orbits of $i$ and $e^{2\pi i/3}$. Let $\mathbb Z$ act on the product $\mathbb C\times\mathbb H^\circ$ by $$(m,n)\cdot(z,\tau) := (z + m\tau + n,\tau)$$ The quotient $\mathbb Z^2\backslash(\mathbb C\times\mathbb H^\circ)$ is an elliptic curve over $\mathbb H^\circ$. Let $SL_2(\mathbb Z)$ act on $\mathbb C\times\mathbb H^\circ$ 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 $SL_2(\mathbb Z)$ on $\mathbb Z^2\backslash(\mathbb C\times\mathbb H^\circ)$, 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 $E_1$?