The one-prime coincidence conjecture for vanishing elliptic-curve constants

Let $\mathcal{E}$ be a non-CM elliptic curve over $\mathbb{Q}$, let $j$ be a positive integer, and let $\mathcal{C}_{\mathcal{E},j}$ denote the constant considered in the paper. A $p$-coincidence is an equality of division fields

$$\mathbb{Q}(\mathcal{E}[j])=\mathbb{Q}(\mathcal{E}[pj])$$

for a prime $p$. The one-prime coincidence conjecture. If $\mathcal{C}_{\mathcal{E},j}=0$, then there exists a prime $p$ such that

$$\mathbb{Q}(\mathcal{E}[j])=\mathbb{Q}(\mathcal{E}[pj]).$$

The preceding corollary proves the converse implication: any such coincidence forces $\mathcal{C}_{\mathcal{E},j}=0$. Thus the conjecture predicts that, for non-CM elliptic curves, all vanishing cases arise from a one-prime coincidence; the statement is based on numerical evidence.