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Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ ofthe finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ ofan elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T.Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curvesums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} \tau(p + a) \simC x$. In this paper, we present a comprehensive study of the constants $C(E)$emerging in the asymptotic study of these elliptic curve divisor sums.Specifically, by analyzing the division fields of an elliptic curve$E/\mathbb{Q}$, we prove upper bounds for the constants $C(E)$ and, in thegeneric case of a Serre curve, we prove explicit closed formulae for $C(E)$amenable to concrete computations. Moreover, we compute the moments of theconstants $C(E)$ over two-parameter families of elliptic curves $E/\mathbb{Q}$.Our methods and results complement recent studies of average constantsoccurring in other conjectures about reductions of elliptic curves byaddressing not only the average behaviour, but also the individual behaviour ofthese constants, and by providing explicit tools towards the computationalverifications of the expected asymptotics.

Constants in Titchmarsh divisor problems for elliptic curves