Ajai Choudhry showed that special cases of the elliptic curve,

$$x(x+a^2)(x+b^2)=y^2\tag1$$

can be used to prove that,

$$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$

has an infinite number of primitive integer solutions. Let $a,b$ be positive integers. Define non-torsion points to $(1)$ of forms,

$$x_i=\Bigl(\frac{p\,\sqrt{a}}{q}\Bigr)^2,\quad\text{and}\quad x_k =\Bigl(\frac{r\,\sqrt{b}}{s}\Bigr)^2\tag3$$

for positive integer $p,q,r,s$.

*Questions:*

- If $(1)$ has a solution of form $x_i$, does it also imply it has for $x_k$?
- If no, what are the conditions on $a,b$ such that $(1)$ has both $x_i,\, x_k$? (Especially for the case $-a+b = 1\;\text{or}\;2$.)

For example, the curve with $a,b = 5,7$ has both,

$$x_1 = \Bigl(\frac{7\cdot71\sqrt{5}}{361}\Bigr)^2,\quad x_2 = \Bigl(\frac{5\cdot47\sqrt{7}}{337}\Bigr)^2 $$

However, the curve with $a,b = 2^7\mp1 =127,129$ that inspired this post only has known,

$$x_2 = \Bigl(\frac{r\sqrt{129}}{s}\Bigr)^2 $$

so I was wondering if it really does not have, or the solutions $p,q$ with $a=127$ to $(3)$ just enormous.