The rank of 2-Selmer group associate to $\theta$-congruent numbers

We study the parity of rank of $2$-$\mathrm{S}\mathrm{e}\mathrm{l}\mathrm{m}\mathrm{e}\mathrm{r}$ groups associated to $\pi \mathrm{/}3$ and $2\pi \mathrm{/}3$-congruent numbers. Our second result gives some positive densities about $\pi \mathrm{/}3$ and $2\pi \mathrm{/}3$ non-congruent numbers which can support the even part of Goldfeld’s conjecture. We give some necessary conditions such that $n$ is non $\pi \mathrm{/}3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp\mathrm{.}11)(\mathrm{m}\mathrm{o}\mathrm{d}24)$, the density of non $\pi \mathrm{/}3$($resp\mathrm{.}$ $2\pi \mathrm{/}3$)-congruent numbers is at least 75%, where $p,q$ are primes.