I want to present a brief question.
I'm curious whether there is any class of Weierstrass form $y^2 = x^3+ax+b$ that we can assign them as rank $0$ by some particular property.
In other words, is there any local, if not global, condition known to $\{(a,b)\ \mid a\in \mathbb{Q}, b\in \mathbb{Q}\}$ that $y^2=x^3+ax+b$ being rank $0$?
As long as I know the paper from Bhargava et al. shows the biggest result regarding the rank of elliptic curves and the related conjectures so far, but I'm not sure whether there is any indication that my curiosity could be solved.
If my question was not clear please let me clarify them with further comments or edits.
Thanks.
Proposition X.6.2(c) of Silverman's The Arithmetic of Elliptic Curves gives one such collection of $a$ and $b$.
Proposition. Let $p$ be an odd prime, and let $E$ be the elliptic curve $$ E: y^2 = x^3 + px \, . $$ Then $$ \operatorname{rank} E(\mathbb{Q}) + \dim_{\mathbb{F}_2} \operatorname{Sha}(E/\mathbb{Q})[2] = \begin{cases} 0 & \text{if $p \equiv 7,11 \bmod{16}$},\\ 1 & \text{if $p \equiv 3,5,13,15 \bmod{16}$},\\ 2 & \text{if $p \equiv 1,9 \bmod{16}$}, \end{cases} $$ where $\operatorname{Sha}(E/\mathbb{Q})$ is the Tate-Shafarevich group. In particular, we see that $a = 0$ and $b = p$ a prime congruent to $7,11 \bmod{16}$ yields an elliptic curve of rank $0$.
See this MO post for a special case of your question.