I've been looking at curves of the form $y^2=x^3+k$ (where k is 6th power free and not divisible by 3^3) and I've noticed that there seems to be distinct grouping in residues classes modulo 504.

One effect that I noticed, in these residue classes, was that the populations of positive and negative k values seem to invert based on odd or even rank.

For example at k = 17 mod 504, even rank positive k values outnumber negative k values whereas it inverts for odd ranks. The following shows rank followed by counts for positive and negative k for this condition.

• Rank 8 [12529, 749]
• Rank 9 [398, 1904]
• Rank 10 [78, 2]
• Rank 11 [0, 4]
• Rank 12 [2, 0]

I would be extremely grateful if someone could help me with an explanation as to why the modulo 504 effect is so pronounced and why the population of positive and negative k inverts for odd and even ranks.

It's probably worth noting at this stage that I am working with an initial population of about 260,000 curves from rank 8 to 12.

Kevin.