Here is some experimental data.

For positive integer $k$ let $E_k:  y^2=x^3+k x $ and $k_1=2,k_2=3$.

According to computations with sage, for $0 < k < 2000$:


1. At least one of $\displaystyle r_{\text{an}}(E_k), r_{\text{an}}(E_k^{(2)}), r_{\text{an}}(E_k^{(3)}), r_{\text{an}}(E_k^{(6)})$ is positive.

2. At least one of $\displaystyle r_{\text{an}}(E_k), r_{\text{an}}(E_k^{(2)}), r_{\text{an}}(E_k^{(3)}), r_{\text{an}}(E_k^{(6)})$ is odd.
3. For $0 < k < 10^5$ at least one of the root numbers of $\displaystyle r_{\text{an}}(E_k), r_{\text{an}}(E_k^{(2)}), r_{\text{an}}(E_k^{(3)}), r_{\text{an}}(E_k^{(6)})$ is $-1$.



Here is [sage code](https://sagecell.sagemath.org/?z=eJw9jlEKwjAMQP8Hu0NQhFaKzAl-CPsQGR5iTKmuauiWlWwVvL3dKoZAQvJeyPLox1fPcDY9PxHOnpAGi2nSYldsswyW3jnDcOs9NfAIpE2TuQASsKanEbkKsDykCYQosajKtkU34v3k-W1EZdfdJVdZLWe_m8Rqq3K1U_u6jhbrYG006fYTvGu4a0XEywkv8c-5osJVPq8wvvBbOUYahVWsQzoZh3oYDI-g6SNYy1CbXz8B0VgYahbyC-QjT3I=&lang=sage&interacts=eJyLjgUAARUAuQ==)

```
#Author Georgi Guninski, Mon Aug 12 04:42:25 PM UTC 2024
lim=100 #upper bound for k
for k in range(2,lim):
    Ei=[EllipticCurve([k*m^2,0]) for m in [1,2,3,6]]
    ra=[E.analytic_rank() for E in Ei]
    rap=[i%2 for i in ra]
    print(k,ra,rap)
    assert any(ra) and any(rap)
print("end")
```