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 $\ell \equiv 7 \pmod{9}$, we show that for $3\ell$ to be a sum of two rational cubes, it is necessary that the ideal class group of $\Q(\sqrt[3]{12\ell})$ contains $\frac{\Z}{6\Z}\oplus \frac{\Z}{3\Z}$ as a subgroup. Moreover, for a positive proportion of primes $\ell \equiv 7 \pmod{9}$, $3\ell$ 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.