On the root numbers of quadruples of quadratic twists of elliptic curves

Question

We got strong numerical evidence for the root numbers and analytic ranks of quadruples of elliptic curves over the rationals.

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:

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

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)