  We consider the loci of curves of genus 2 and 3 admitting a $d$-to-1 map to a
genus 1 curve. After compactifying these loci via admissible covers, we obtain
formulas for their Chow classes, recovering results of Faber-Pagani and van
Zelm when $d=2$. The answers exhibit quasimodularity properties similar to
those in the Gromov-Witten theory of a fixed genus 1 curve; we conjecture that
the quasimodularity persists in higher genus, and indicate a number of possible
variants.
