2 added motivation - shows genus 1 curves cannot have fine moduli space

(Edit: added motivation for my "answer" even though it isn't exactly what was asked for. It shows the moduli space of genus 1 curves cannot be fine - see last paragraph.)

The Hopf surface is a (non-algebraic) example of a non-trivial family of isomorphic elliptic curves. (Note: here Here I am using use "elliptic curve" to mean "smooth complex curve of genus 1", which is probably not what you mean. (In particular , my "elliptic curves " don't come with a have no distinguished base point. I concede this is probably not what you want, but it is a nice example to see anyway.point.)

The Hopf surface $X$ is a quotient of $\mathbb{C}^2\setminus0$ by the action of $\mathbb Z$ generated by $z \mapsto 2z$. Since this action commutes with the action of $\mathbb{C}^*$ there is a map $X \to \mathbb{CP}^1$. Each fibre is a copy of $\mathbb{C}\setminus{0}$ divided by the action $z\sim 2z$. So all the fibres are isomorphic elliptic curves. (If you want base points on each curve, this example doesn't work because the fibration $X \to \mathbb{CP}^1$ doesn't have a section.) On the other hand, its easy to see that $X$ is diffeomorphic to $S^1\times S^3$ and so you can't trivialise the fibration even topologically. Replacing $z\mapsto 2z$ by $z\mapsto \lambda z$ for $\lambda\in \mathbb{C}$ non-zero, I guess you can get any genus-1 curve to appear as all fibres of a topologically non-trivial family.

Even though this is not eaxctly what you're looking for, I thought it worth giving the example anyway, because it's a very simple way to see that the moduli space of genus 1 curves cannot be fine. (Fine moduli spaces carry a universal family and all other families are pulled back from the universal family via the map to moduli space - in particular a family of isomorphic objects in such a moduli space is pulled back by the constant map and so must be trivial. One often sees that objects cannot be parametrised by a fine moduli space by giving examples of special objects with "extra" automorphisms. The Hopf surface is an alternative to that approach in this situation.)

1

The Hopf surface is a (non-algebraic) example of a non-trivial family of isomorphic elliptic curves. (Note: here I am using "elliptic curve" to mean "smooth complex curve of genus 1". In particular, my "elliptic curves" don't come with a distinguished base point. I concede this is probably not what you want, but it is a nice example to see anyway.)

The Hopf surface $X$ is a quotient of $\mathbb{C}^2\setminus0$ by the action of $\mathbb Z$ generated by $z \mapsto 2z$. Since this action commutes with the action of $\mathbb{C}^*$ there is a map $X \to \mathbb{CP}^1$. Each fibre is a copy of $\mathbb{C}\setminus{0}$ divided by the action $z\sim 2z$. So all the fibres are isomorphic elliptic curves. (If you want base points on each curve, this example doesn't work because the fibration $X \to \mathbb{CP}^1$ doesn't have a section.)

On the other hand, its easy to see that $X$ is diffeomorphic to $S^1\times S^3$ and so you can't trivialise the fibration even topologically.