Watkins's conjecture for elliptic curves over the rationals

Let $E/\mathbb{Q}$ be an elliptic curve. Its modular degree $m_E$ is the minimal degree of a surjective modular parametrization $X_0(N)\to E$ defined over $\mathbb{Q}$ and sending the cusp at infinity to the identity of $E$. Write $\nu_2(m_E)$ for the $2$-adic valuation of $m_E$. Watkins's conjecture. For every elliptic curve $E/\mathbb{Q}$,

$$\operatorname{rank}_{\mathbb{Z}} E(\mathbb{Q})\leq \nu_2(m_E).$$

The conjecture concerns an upper bound for the Mordell–Weil rank in terms of the modular degree. It has been established on average, and the paper studies it in thin families, including elliptic curves with rational torsion; its general status remains open.