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For positive integer $b$ define the curve $C_b : y^2=x^4+x+b^2$. $C_b$ is genus one and has the rational points: $(0,\pm b),(-1,\pm b)$ and one more point from the reciprocal of the polynomial y=0

Let $r(b)$ denote the rank of the Jacobian of $C_b$.

We are interested in the distribution of $r(b)$ as $b$ varies.

Here is some experimental data from sagemath for $1 \le b \le 857$.

rank: all curves

2: 1, 3: 187, 4: 346, 5: 235, 6: 75, 7: 13

rank:percent of curves

2: 0, 3: 21, 4: 40, 5: 27, 6: 8, 7: 1

As an aside, we are surprised we got 13 curves of rank 7 with so simple construction.

Q1 What can we say in general about the distribution of $r(b)$?

Q2 For which $r$ the curves with $r(b)=r$ are of positive density?

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This is only a partial answer, but it explains some features of the observed data. A model of the Jacobian of $C_b$ is given by $$E_b : y^2 + 2y = x^3 - 4b^2 x.$$ The Abel–Jacobi map with base point $(1 : 1 : 0)$ (at infinity, in projective coordinates) is then given by $$(x, y) \mapsto (2x^2 + 2y, 4x^3 + 4xy).$$ Under this map, the known rational points on $C_b$ map to the following points of $E_b$: $$\{(0, -2), (2b, 0), (-2b, 0), (2b + 2, -4b - 4), (-2b + 2, 4b - 4)\}$$ We compute the height pairing matrix (looks like Sage hasn't implemented this for function fields yet, but Magma has) and find that these points generate a rank 3 subgroup with basis $$\{(0, -2), (2b, 0), (2b + 2, -4b - 4)\}.$$

From this, we can conclude that the elliptic curve $E_b$ has rank at least 3 for all values of $b$ for which these three points remain independent. This explains why almost all the curves you computed have rank at least 3.

We also have a conjectural explanation of the appearance of many rank 4 curves. Assuming that all elliptic curves over $\mathbb{Q}$ have finite Tate–Shafarevich group, the parity conjecture is known to hold, meaning that the analytic ranks are equidistributed modulo 2. Assuming the BSD conjecture, this means that the same is true of algebraic ranks. If the same distribution holds in the family of elliptic curves $E_b$, then at least half of the curves in the family have rank 4 or higher.

But why are there so many curves of rank 5? It could just be a small-numbers effect and these curves will become increasingly sparse as we look at larger and larger $b$. Or there could be some other phenomenon causing it—say, something like a fourth independent point that only occurs for some values of $b$, perhaps depending on a congruence condition, which could bump the minimum rank up to 4 within some subfamily (and then the parity conjecture would imply that existence of many curves in this family of rank 5 or more). I briefly looked at the data and didn't see this sort of simple explanation for the rank 5 curves, but it's a possibility.

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    $\begingroup$ A more likely reason for the rank $5+$ curves is that there is some rational function $f(c)\in\mathbb Q(c)$ such that $$E_{f(c)}:y^2+2y=x^3-4f(c)^2x$$ has rank 4 over $\mathbb Q(c)$. In more geometric language, $E_b\to\mathbb P^1_{\mathbb Q}$ is an elliptic surface of rank 3, and maybe there is a finite cover $\mathbb P^1_{\mathbb Q}\to \mathbb P^1_{\mathbb Q}$ so that the pull-back elliptic surface acquires one or more new independent sections. One way to search for $f(c)$ is to assume it has small degree, use the $b_i$ values with high rank, and solve $f(c)=b_i$ for the coeffs of $f$. $\endgroup$
    – Joe Silverman
    Commented Feb 16, 2021 at 2:22
  • $\begingroup$ @JoeSilverman Experimentally we have rank 5 with probability about 27%. Are you trying to explain this with rational function equal square with probability about 27%? $\endgroup$
    – joro
    Commented Feb 16, 2021 at 14:41
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    $\begingroup$ @joro I'm just suggesting a line of attack that might be worth checking. But there is also definitely a "small number" phenomenon that shows up in experiments involving ranks in families, even with much larger search spaces than you've used. Indeed, there are often error terms like $O(1/\log|b|)$ if the family is parametrized by $b$. So checking $|b|\le1000$ can be mildly suggestive, but that $27\%$ you get could become (say) $15\%$ if you went to $|b|\le10000$ and $9\%$ if you went to $|b|\le100000$, and it would still really not help in guessing whether the percentage goes to~$0$. $\endgroup$
    – Joe Silverman
    Commented Feb 16, 2021 at 15:50
  • $\begingroup$ Thanks. sage and mwrank are slow in computing the rank for large $b$, is there a CAS which can compute the rank for say 10 b's for $b$ as large as possible? $\endgroup$
    – joro
    Commented Feb 16, 2021 at 16:01
  • $\begingroup$ How is the finiteness of Sha relevant to equidistribution of the parity? $\endgroup$
    – Will Sawin
    Commented Apr 7, 2021 at 14:42

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