Are morphism from $\mathbb{P}^{1}$ to itself often liftable to a morphism from a curve to an elliptic curve with bounded degree?

Question

(all morphism here means birational)

(the ground field is "small", but I don't think it should matter)

Here is the picture. I have a morphism $f:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$. I want to lift this up to a morphism $\tilde{f}:C\rightarrow E$ where $C$ is a curve and $E$ is an elliptic curve with projection $\pi_{C}:C\rightarrow\mathbb{P}^{1}$ and $\pi_{E}:E\rightarrow\mathbb{P}^{1}$, so that we have a commutative diagram $f\circ\pi_{C}=\pi_{E}\circ\tilde{f}$. Now, most of these are not fixed. A few things are fixed: the degree of $\pi_{C}$ is fixed at some numbers $d$ independent of $f$, the degree of $\pi_{E}$ is fixed at 2, but other than that $C,E,\pi_{C},\pi{E},\tilde{f}$ we are allowed to freely choose dependent on $f$.

So my actual question is: is there any $d$ in which this lifting can be done "often", in the sense that it is non-special, ie. failure to lift is not a generic property? Or is this just always false?

I previously tried this for $d=2$, $C$ being (another) elliptic curve, and $\pi_{C}$ be the x-coordinate, but this gives a strong restriction on $f$, that is ramification points of $f$ must lift to $2n$-torsion points on $C$, and thus knowing $2$ points gives us at most $4n^{2}$ possible elliptic curves.

Answer 1

$\newcommand{\CC}{\mathbb C}$If there is a commutative diagram as requested, then there is also one with $C$ and $\pi_C$ replaced by $C'$ and $\pi_{C'}$, where $\pi_{C'}$ has degree $2$ by letting $C'$ be a connected component of the fiber product of $f$ and $\pi_E$.

Or in terms of function fields, with $x$ a transcendental over $\CC$ and $t=f(x)$: The fields $\CC(x)$ and $\CC(E)$ are intermediate fields of $\CC(t)$ and $\CC(C)$. Now define $C'$ by the new intermediate field $\CC(C')=\CC(x, E)$. Note that $[\CC(C'):\CC(x)]=2$, so $\pi_{C'}$ has degree $2$.

So one is reduced to the case $d=2$. And there is always a cover as requested: Start with any $E$ and $\pi_E$, let $L$ be an algebraic extension of $\CC(t)$ which contains $\CC(x)$ and $\CC(E)$, and define $C$ by $\CC(C)=\CC(x,E)\subseteq L$.

But for generic functions $f$, like Morse functions, the genus of $C$ goes to infinity which $\operatorname{deg} f$. So there is probably little use of the lift $\tilde f$.