Consider the moduli space $$ \overline{M}_{0,4}(B\mathbb{Z}/2) $$ This has virtual (and real) dimension one. In a certain sense this moduli space paramaterizes "genus 1 hyperelliptic curves"; that is, given a family of objects in this moduli space, by pulling back the universal family $pt \to B\mathbb{Z}/2$ we obtain a genus one curve with marked 2-torsion points. In particular, there is a morphism $$ \overline{M}_{0,4}(B\mathbb{Z}/2) \to \overline{M}_{1,4} $$ whose image is those genus 1 curves whose four marked points are exactly the 2-torsion points.

Because of this map, we can pull back the Hodge bundle to $\overline{M}_{0,4}(B\mathbb{Z}/2)$, and so we can consider the integral $$ \int_{\overline{M}_{0,4}(B\mathbb{Z}/2)} \lambda_1 $$

**Question** What is the value of this integral?

Some thoughts: We know that $\int_{\overline{M}_{1,1}} \lambda_1 = \frac{1}{24}$. Since the Hodge bundle pulls back via the forgetful maps (forgetting marked points), it seems that we should end up with *either* $3!\frac{1}{24} = \frac{1}{4}$ (choosing our favourite marked point, mapping to $\overline{M}_{1,1}$) or $4!\frac{1}{24} = 1$ (forgetting all the marked points).

I suppose another way of phrasing this: What is the degree of the map $\overline{M}_{0,4}(B\mathbb{Z}/2) \to \overline{M}_{1,1}$?