Dear MO Community,

this is a pretty vague title, so let me tell you the precise observation I have made.

Consider the family of elliptic curves over $\mathbf{Q}$ having a rational $5$-torsion point $P$. They are given by $$E_d: Y^2 + (d+1)XY +dY=X^3+dX^2,$$ for $d \in \mathbf{Q}^*$ and $P=(0,0)$. Let $\eta: E_d \rightarrow E_d'$ the isogeny whose kernel consists exactly of the five rational $5$-torsion points.

Now assume that $E_d$ has rank $1$. After modding out torsion the Mordell-Weil group is isomorphic to $\mathbf{Z}$, and hence, $\eta$ induces an injective group homomorphism $\mathbf{Z} \rightarrow \mathbf{Z}$ which is either an isomorphism or has cokernel of size $5$.

It seems to me that this map tends to be an isomorphism.

To be precise, among all $d$, such that the numerator and denominator is bounded by $100$, there are $3,038$ elliptic curves of analytic rank equal to $1$ (out of $6,087$ total curves), and among those the above map $\mathbf{Z} \rightarrow \mathbf{Z}$ is an isomorphism in $91.2\%$ of the cases.

So I wonder, what you should expect on average? $50\%$? $100\%$?

Maybe, this was just a coincidence in a small database. Maybe someone has seen a similar behaviour somewhere else. Maybe this is nothing new and I just haven't heard about it. I am curious to read what you think about it.

Many thanks.

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