Related to this question.

Let $k,k_1,k_2$ be squarefree pairwise coprime integers.

Assume $|k|>1$ and $|k_1|>1$ and $|k_2|>1$.

Define the elliptic curves over the rationals:

$$E_0:x^3+k x=y^2, E_1: x^3+k k_1^2 x,E_2: x^3+k k_2^2=y^2, E_3: x^3+k k_1^2 k_2^2=y^2$$

If necessary, assume widely believed conjectures like BSD, Parity, GRH.

Which of the following conjectures are true:

(1) The root number of at least one of the $E_i$ is $-1$.

(2) The analytic rank of at least one of the $E_i$ is odd.

(3) The analytic rank of at least one of the $E_i$ is positive.

For $k_1=2,k_2=3$, all of the above hold up to $k=2000$ and reportedly Elkies verified stronger claim to $10^4$.

Here is sage code in case someone want to verify

#Author Georgi Guninski, Tue Aug 13 02:08:24 PM UTC 2024
lim=100 #upper bound for k
def BSD_r00t(lim,k_1,k_2):
    #returns True if all conjectures pass, else (False,"reason")
    assert abs(k_1)>1 and abs(k_2)>1,"luser error |k_i|=1"
    assert gcd(k_1,k_2)==1,"luser error  coprime k_1,k_2"
    assert ZZ(k_1).is_squarefree(),"luser error k_1 squarefree"
    assert ZZ(k_2).is_squarefree(),"luser error k_2 squarefree"
    for k in range(2,lim):
        if gcd(k,k_1*k_2) != 1:  continue
        Ei=[EllipticCurve([k*m^2,0]) for m in [1,k_1,k_2,k_1*k_2]]
        r00t=[E.root_number() for E in Ei]
        ran=[E.analytic_rank() for E in Ei]
        ranmod2=[i%2 for i in ran]
        print(k,ran,ranmod2,r00t)
        if not -1 in r00t:  return (False,"r00t")
        if max(ran)==0:  return (False,"analytic rank")
        if max(ranmod2)==0:  return (False,"parity")
    return True,[]
res=BSD_r00t(lim,2,3)
print("result=",res)