Start with the product $\mathbb{C}\times\mathcal{H}$ ($\mathcal{H} = $ upper half plane). Define an action of $\mathbb{Z}^2$ on the left by $(m,n)\cdot(z,\tau) := (z + m\tau + n,\tau)$.
Then the quotient $\mathbb{Z}^2\backslash(\mathbb{C}\times\mathcal{H})$ is a family of elliptic curves over $\mathcal{H}$. Since $\mathcal{H}$ is simply connected, this should be topologically a trivial fibration (though not a complex-analytically trivial fibration?)
Consider the left action of SL($2,\mathbb{Z}$) on $\mathbb{C}\times\mathcal{H}$ as follows: given, $\gamma = [[a,b],[c,d]]\in\text{SL}(2,\mathbb{Z})$ (dunno how else to write matrices here...), define $\gamma\cdot(z,\tau) := \left(\frac{z}{c\tau + d}, \frac{a\tau + b}{c\tau + d}\right)$. Let $\mathcal{H}^\circ$ denote the upper half plane minus the SL($2,\mathbb{Z}$)-orbits of $i$ and $e^{2\pi i/3}$. This action then gives an action of $\text{SL}(2,\mathbb{Z})$ on $\mathbb{Z}^2\backslash(\mathbb{C}\times\mathcal{H}^\circ)$, and the quotient is an elliptic family over $\text{SL}(2,\mathbb{Z})\backslash\mathcal{H}^\circ$ (ie, the $j$-line with 0, 1728 removed).
Is this family an algebraic family of elliptic curves? (I know there's a commonly used example of an algebraic family over the $j$-line minus 0, 1728 where the fiber over $j$ has $j$-invariant $j$...is this it?)
My real question though, is...does this family have a nontrivial (complex-analytic) section (other than the identity)? Does anyone know the equation for this family?
thanks