$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\AA}{\mathbb{A}}$
$\newcommand{\hH}{\mathcal{H}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\SL}{\text{SL}}$

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 $E_1 := y^2 + xy  = x^3 - \frac{36}{j-1728} - \frac{1}{j-1728}$.

On the other hand, one can also come up with an analytical/topological description of a family with the same property. 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 acting on $\hH^\circ$ in the usual manner. This action descends to an action of $\SL_2(\ZZ)$ on $\ZZ^2\backslash(\CC\times\hH^\circ)$, and the quotient $\SL_2(\ZZ)\backslash(\ZZ^2\backslash(\CC\times\hH^\circ))$ is an elliptic curve $E_2$ over $\SL_2(\ZZ)\backslash\hH^\circ$ (ie, the $j$-line punctured at 0,1728).

My questions are: 

1. Are $E_1$ and $E_2$ isomorphic? (is the equation given above an equation for $E_2$?)
2. If they are, is $E_1$ or $E_2$ the unique family over $\AA^1_\CC\setminus\{0,1728\}$ whose fiber above $j$ has $j$-invariant $j$?
3. If they aren't, has $E_2$ been studied at all? Is there a name for this family? Does anyone know where I can find an equation for it?
4. Does $E_2$ have a nontrivial section? (ie, other than the identity section).