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For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros 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.

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor