Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ as the set of (isomorphism classes of) elliptic curves $E$ over $\mathbf{Q}$ that admit a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$.

Question 1. Let $E$ be in $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ and choose a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$. Let $X$ be the analytification of $E_\mathbf{C}$. There exists a $\tau$ in the complex upper half plane such that $X = \mathbf{C}/\mathbf{Z}+\tau\mathbf{Z}$. Can we choose $\tau$ using the data $(b_1,b_2,\ldots,b_r,d,f)$?

Question 2. It follows from Faltings's theorem that the set $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite. Is there a more elementary proof of this? (One could look for a bound on the height of the j-invariant for example. This is related in a sense to the first question)

Example. We take $r=3$ and $(b_1,b_2,b_3) = (0,1,\infty)$. So $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite by (for example) the theory of dessins d'enfants. For $d=2$ this set is empty. For $d=3$ it consists of the (isomorphism class of) the elliptic curve with j-invariant 0. For $d=4$ we get the elliptic curves with $j=1728$ and $j=207647/6561$ (and the one with $j=0$, I believe).

Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ as the set of (isomorphism classes of) elliptic curves $E$ over $\mathbf{Q}$ that admit a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$.

Question 1. Let $E$ be in $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ and choose a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$. Let $X$ be the analytification of $E_\mathbf{C}$. There exists a $\tau$ in the complex upper half plane such that $X = \mathbf{C}/\mathbf{Z}+\tau\mathbf{Z}$. Can we choose $\tau$ (or $q=e^{2\pi i \tau}$) using the data $(b_1,b_2,\ldots,b_r,d,f)$?

Question 2. It follows from Faltings's theorem that the set $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite. Is there a more elementary proof of this?

EDIT: Let me describe how the elliptic curve is given (in the set-up I have in mind).

Let $U$ be an open subscheme of $\mathbf{P}^1_\mathbf{Z}$ with complement $D$. We suppose that the closed subscheme $D$ is a horizontal divisor on $\mathbf{P}^1_\mathbf{Z}$ such that the base change $D_\mathbf{Q}$ equals $B$ defined above. Let $V\longrightarrow U$ be a finite etale morphism, with $V$ connected. Let $g:Y\longrightarrow \mathbf{P}^1_\mathbf{Q}$ be the normalization of $\mathbf{P}^1_\mathbf{Q}$ in the function field of $V$. We make the following extra assumptions:

1. $Y$ has a $\mathbf{Q}$-rational point.

2. The genus of $Y$ equals 1.

So the morphism $f$ arises like this.

I'm actually more interested in the set-up described above without assumptions 1 and 2. I just figured it would be an easy case to start with because it could/should be handled more directly.

Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ as the set of (isomorphism classes of) elliptic curves $E$ over $\mathbf{Q}$ that admit a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$.

Question 1. Let $E$ be in $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ and choose a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$. Let $X$ be the analytification of $E_\mathbf{C}$. There exists a $\tau$ in the complex upper half plane such that $X = \mathbf{C}/\mathbf{Z}+\tau\mathbf{Z}$. Can we choose $\tau$ using the data $(b_1,b_2,\ldots,b_r,d,f)$?

Question 2. It follows from Faltings's theorem that the set $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite. Is there a more elementary proof of this? (One could look for a bound on the height of the j-invariant for example. This is related in a sense to the first question)

Question 3. Can we bound the number of elements of $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$?

Question 4. What is the precise connection with the Hurwitz space?

Question 5. Can one give an explicit description of $$\textrm{Ell}(b_1,b_2,\ldots,b_r) := \bigcup_{d=3}^\infty \textrm{Ell}(b_1,b_2,\ldots,b_r,d) ? $$

Example. We take $r=3$ and $(b_1,b_2,b_3) = (0,1,\infty)$. So $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite by (for example) the theory of dessins d'enfants. For $d=2$ this set is empty. For $d=3$ it consists of the (isomorphism class of) the elliptic curve with j-invariant 0. For $d=4$ we get the elliptic curves with $j=1728$ and $j=207647/6561$ (and the one with $j=0$, I believe). By Belyi's theorem, we have that $$\textrm{Ell}(b_1,b_2,\ldots,b_r) := \bigcup_{d=3}^\infty \textrm{Ell}(b_1,b_2,\ldots,b_r,d) $$ is the $j$-line.

Example. We take $r=4$ and $(b_1,b_2,b_3,b_4) = (0,1,\infty,s)$. For $d=2$ you get at least one elliptic curve. (The one with equation $y^2 = x(x-1)(x-s)$.)

Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ as the set of (isomorphism classes of) elliptic curves $E$ over $\mathbf{Q}$ that admit a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$.

Question 1. Let $E$ be in $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ and choose a finite morphism $f:E\longrightarrow \mathbf{P}^1_\mathbf{Q}$ of degree $d$ which is etale outside $\{b_1,b_2,\ldots,b_r\} \subset \mathbf{P}^1(\mathbf{Q})$. Let $X$ be the analytification of $E_\mathbf{C}$. There exists a $\tau$ in the complex upper half plane such that $X = \mathbf{C}/\mathbf{Z}+\tau\mathbf{Z}$. Can we choose $\tau$ using the data $(b_1,b_2,\ldots,b_r,d,f)$?

Question 2. It follows from Faltings's theorem that the set $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite. Is there a more elementary proof of this? (One could look for a bound on the height of the j-invariant for example. This is related in a sense to the first question)

Example. We take $r=3$ and $(b_1,b_2,b_3) = (0,1,\infty)$. So $\textrm{Ell}(b_1,b_2,\ldots,b_r,d)$ is finite by (for example) the theory of dessins d'enfants. For $d=2$ this set is empty. For $d=3$ it consists of the (isomorphism class of) the elliptic curve with j-invariant 0. For $d=4$ we get the elliptic curves with $j=1728$ and $j=207647/6561$ (and the one with $j=0$, I believe).