  We study Severi curves parametrizing rational bisections of elliptic
fibrations associated to general pencils of plane cubics. Our main results show
that these Severi curves are connected and reduced, and we give an upper bound
on their geometric genus using quasi-modular forms. We conjecture that these
Severi curves are eventually reducible, and we formulate a precise conjecture
for their degrees in $\mathbb{P}^2$, featuring a divisor sum formula for
collision multiplicities of branch points.
