Let $S$ be the set of isomorphism classes of elliptic curves over $\mathbb{Q}$. Consider the following claim.
There is a map $f:S\to \mathbb{N}$ such that $|f^{-1}(n)|$ is finite for all $n\in \mathbb{N}$. Moreover $|f^{-1}(n)|$ assumes infinitely many values ranging over $n\in \mathbb{N}$. There are also curves $C_n$ such that given $n\in \mathbb{N}$ there are non-constant maps from $C_n$ to all curves in $f^{-1}(n)$. Finally the genus of $C_n$ is bounded polynomially in $|f^{-1}(n)|$.
At first I thought this is true with $f$ the conductor and $C_n=X_1(n)$ but it's not (consider the cubes of primes for example).
Is it true for some other $f$ and $C_n$?
