On certain root number $1$ cases of the cube sum problem

We consider certain families of integers $n$ determined by some congruence condition, such that the global root number of the elliptic curve $E_{-432n^2}:Y^2=X^3-432n^2$ is $1$ for every $n$, however a given $n$ may or may not be a sum of two rational cubes. We give explicit criteria in terms of the $2$-parts and $3$-parts of the ideal class groups of certain cubic number fields to determine whether such an $n$ is a cube sum. In particular, we study integers $n$ divisible by $3$ such that the global root number of $E_{-432n^2}$ is $1$. For example, for a prime $\mathrm{\ell }\equiv 7(\mathrm{m}\mathrm{o}\mathrm{d}9)$, we show that for $3\mathrm{\ell }$ to be a sum of two rational cubes, it is necessary that the ideal class group of $\text{Q}(\sqrt[3]{12\mathrm{\ell }})$ contains $\frac{\mathbb{Z}}{6\mathbb{Z}}\oplus \frac{\mathbb{Z}}{3\mathbb{Z}}$ as a subgroup. Moreover, for a positive proportion of primes $\mathrm{\ell }\equiv 7(\mathrm{m}\mathrm{o}\mathrm{d}9)$, $3\mathrm{\ell }$ can not be a sum of two rational cubes. A key ingredient in the proof is to explore the relation between the $2$-Selmer group and the $3$-isogeny Selmer group of $E_{-432n^2}$ with the ideal class groups of appropriate cubic number fields.