I was playing around with *sage*, when I found that the Mordell-Weil 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, so why would their ranks be correlated?
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* primes $p>2$.
I verified this using the following code, written in *sage*:
<blockquote>
for p in primes(100):<br>
E1 = EllipticCurve(QQ,[0,p^3])<br>
E2 = EllipticCurve(QQ,[0,-p^3])<br>
print("p = "+QQ(p).str()+":"),<br>
rank1 = E1.selmer_rank()<br>
rank2 = E2.selmer_rank()<br>
print([rank1,rank2])<br>
</blockquote>
which gives
<blockquote>
p = 2: [2, 1]
p = 3: [1, 1]
p = 5: [1, 1]
p = 7: [2, 2]
p = 11: [2, 2]<br>
p = 13: [1, 1]
p = 17: [1, 1]
p = 19: [2, 2]
p = 23: [2, 2]
p = 29: [1, 1]<br>
p = 31: [2, 2]
p = 37: [3, 3]
p = 41: [1, 1]
p = 43: [2, 2]
p = 47: [2, 2]<br>
p = 53: [1, 1]
p = 59: [2, 2]
p = 61: [3, 3]
p = 67: [2, 2]
p = 71: [2, 2]<br>
p = 73: [1, 1]
p = 79: [2, 2]
p = 83: [2, 2]
p = 89: [1, 1]
p = 97: [1, 1]
</blockquote>
I have been trying to prove this by following a case distinction according to the residue class of $p$ modulo $12$, and performing a partial $2$-descent for each of the cases, but I keep getting distracted by the thought that
there must be a neater explanation that I'm missing. Hence my question:
<blockquote>
<b>Is there?</b>
</blockquote>