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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 $\displaystyle (E_k), (E_k^{(2)}), (E_k^{(3)}), (E_k^{(6)})$ is $-1$.

Here is sage code

#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")