I was playing around with sage, when I found that the ranks (over $\mathbf{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few exceptions occur at $p=37$, $p=61$, $p=157$, $p=193$, $\ldots$. This pattern struck me as odd, since the two curves are non-isogenous over the ground field.

After some reflection and further experimentation, I found out that if one looks instead at the $2$-Selmer ranks, there is even a stronger pattern: they seem to agree for all $p>2$ (again with $p$ prime).

I verified this using the sage code

for p in primes(100):
     E1 = EllipticCurve(QQ,[0,p^3])
     E2 = EllipticCurve(QQ,[0,-p^3])
     print("p = "+QQ(p).str()+":"),
     rank1 = E1.selmer_rank()
     rank2 = E2.selmer_rank()
     print([rank1,rank2])

which gives

p = 2: [2, 1]  p = 3: [1, 1]   p = 5: [1, 1]   p = 7: [2, 2]   p = 11: [2, 2]
p = 13: [1, 1] p = 17: [1, 1] p = 19: [2, 2] p = 23: [2, 2] p = 29: [1, 1]
p = 31: [2, 2] p = 37: [3, 3] p = 41: [1, 1] p = 43: [2, 2] p = 47: [2, 2]
p = 53: [1, 1] p = 59: [2, 2] p = 61: [3, 3] p = 67: [2, 2] p = 71: [2, 2]
p = 73: [1, 1] p = 79: [2, 2] p = 83: [2, 2] p = 89: [1, 1] p = 97: [1, 1]

I have been trying to prove this by making a case distinction according to the residue class of $p$ modulo $12$, and performing a partial $2$-descent in each of those cases, but I keep getting distracted by the thought that there must be a neater explanation that I'm missing.

Is there?