  For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials
of rational elliptic curves $E$ of conductor $p$. Ono made the observation that
these zeros of are often $j$-invariants of supersingular elliptic curves over
$\overline{\mathbb{F}_p}$. We show that these supersingular zeros are in
bijection with zeros modulo $p$ of an associated quaternionic modular form
$v_E$.
  This allows us to prove that if the root number of $E$ is $-1$ then all
supersingular $j$-invariants of elliptic curves defined over $\mathbb{F}_{p}$
are zeros of the corresponding divisor polynomial.
  If the root number is $1$ we study the discrepancy between rank $0$ and
higher rank elliptic curves, as in the latter case the amount of supersingular
zeros in $\mathbb{F}_p$ seems to be larger. In order to partially explain this
phenomenon, we conjecture that when $E$ has positive rank the values of the
coefficients of $v_E$ corresponding to supersingular elliptic curves defined
over $\mathbb{F}_p$ are even. We prove this conjecture in the case when the
discriminant of $E$ is positive, and obtain several other results that are of
independent interest.
