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.



Parity dependent population inversion in Mordell elliptic curves
The parity of the analytic rank of any elliptic curve $E\_k: y^2 = x^3 + k$
over ${\bf Q}$ was determined in the paper



> 
> Liverance, Eric:
>  A formula for the root number of a family of elliptic curves.
>  *J. Number Theory* **51** #2, 288--305 (1995).
> 
> 
> 


(This reference was posted a few weeks ago by Larry Washington
to the NMBRTHRY mailing list in response to another question on
the parity of some curves $E\_k$.)


Suppose $k$ is a sixth-power-free integer, and write $k = 2^b 3^c k\_1$
with $\gcd(k\_1,6) = 1$. Then Liverance writes the sign of the
functional equation of $E\_k$ as $-w^{\phantom.}\_2 w^{\phantom.}\_3 (-1)^r$, where:


$w^{\phantom.}\_2 = 1$ or $-1$ depending only on $b$ and on $3^c k\_1 \bmod 4$;


$w^{\phantom.}\_3 = 1$ or $-1$ depending only on $c$ and on $2^b k\_1 \bmod 9$;


and $r$ is the number (*without* multiplicity) of prime factors of $k\_1$
congruent to $-1 \bmod 6$.


[NB Liverance's $w^{\phantom.}\_2$ and $w^{\phantom.}\_3$ are not quite the same as
the local root numbers of $E\_k$ at $2$ and $3$, though they are
closely related with these root numbers.]


Therefore: if $k$ is not divisible by the square of any prime
congruent to $-1 \bmod 6$, then the parity is determined entirely by
$w^{\phantom.}\_2$, $w^{\phantom.}\_3$, and whether $k>0$ or $k<0$.
This happens in particular if $k$ has no $-1 \bmod 6$
factors at all, which is the case for the quadratic polynomials
$k = -108 t^2 + 36 t - 7$ and $k = -108 t^2 + 36 t - 67$
in K.Acres' self-answer, because their discriminants are of the form $-3d^2$.
Moreover, for each of these polynomials $k \bmod 36$ is constant with $\gcd(k,6) = 1$,
so the sign of the functional equation is the same for all $t$.
The smaller examples $-6t^2-8$ and $6t^2+2$ from K.Acres' comment
also have discriminants $-3d^2$, but showing that their sign is constant
requires some case analysis for the variation of $b$, $c$, and $k\_1$ with $t$.


Even when the sign is not fully predictable there can be large biases.
For example, in the arithmetic progression $k=36n+1$ the sign is
*usually* $+1$ if $k>0$ and $-1$ if $k<0$. There are exceptions
(starting at $k=36\cdot8+1 = 17^2$ and $k=36\cdot9+1 = 5^2 13$),
but they require that $k$ be divisible by $p^2$ for some
$p \equiv -1 \bmod 6$, and that happens less than 6% of the time.


