There's a natural map $f:\overline{\mathcal{M}}\_{1,1}\to \overline{M}\_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^\*:\mathbb{Z}\to\mathbb{Z}$ is an homomorphism. What homomorphism? My guess is: $x \mapsto 24 x$ since the generator of the stack is the Hodge class, which has degree 1/24. Do you agree?



$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
I believe the number is 12.


I will assume the characteristic of the base field is not 2 or 3 so that I can use $\overline{\mathcal{M}}\_{1,1} \simeq \mathbf{P}(4,6)$. Recall that $\mathbf{P}(4,6)$ is constructed by dividing $V = \mathbf{A}^2 \smallsetminus \{ (0,0) \}$ by the weight $(4,6)$-action of $\mathbf{G}\_m$.


The line bundle $\mathcal{O}(1)$ (which coincides with the Hodge bundle and generates the Picard group) can be constructed as the equivariant line bundle $V \times \mathbf{A}^1$ on $V$ with $\mathbf{G}\_m$ acting by weights $(4,6,1)$. Then $\mathcal{O}(4)$ has a canonical section $g\_4$ and $\mathcal{O}(6)$ has the section $g\_6$. In $\mathcal{O}(12)$ we have the two sections


$1728 g\_4^3$


$\Delta = g\_4^3 - 27 g\_6^2$.


This linear series defines $j : \mathbf{P}(4,6) \rightarrow \mathbf{P}^1$.


Note that $j$ has degree $1/2$ because of the generic automorphism on $\mathbf{P}(4,6)$. That is, $j\_\ast j^\ast$ is multiplication by $1/2$. Thus the Hodge class $\lambda$ satisfies


$\int\_{\overline{\mathcal{M}}\_{1,1}} \lambda = \int\_{\mathbf{P}(4,6)} c\_1(\mathcal{O}(1))
= \frac{1}{12} \int\_{\mathbf{P}(4,6)} c\_1(j^\ast \mathcal{O}(1)) = \frac{1}{12} \int\_{\mathbf{P}^1} j\_\ast j^\ast c\_1(\mathcal{O}(1)) = \frac{1}{24}$.


Another thing that may be confusing here is that $j$ is generically unramified, so that a local equation for a point in $\mathbf{P}^1$ pulls back under $j$ to a local equation in $\overline{\mathcal{M}}\_{1,1}$. Thus $j^\ast \Delta = \delta$ (if $\Delta$ denotes the boundary in the coarse moduli space and $\delta$ the boundary in the stack). 


