Is there a relationship between points on 2 different elliptic curves that share zeros?

Solution 1:

In general, there should not be any relation. If $K$ is a field of characteristic $0$, then any elliptic curve $E/K$ with a nontrivial $2$-torsion point can be written so that $E$ is given by a Weierstrass equation (with $a_1=a_3=0$) and the point being at $(x_0, 0)$ for any choice of $x_0 \in K$.

If two elliptic curves share all their zeroes, then by construction they must be given by Weierstrass equations $y^2 = f(x)$ and $y^2 = df(x) $ for some $d \in K$. This is exactly what it means to be a "quadratic twist". However there is nothing (easy) to say about the relationship between the rational points of quadratic twists.

In your case we have the curves $$E : y^2 = x(x+1)(2x - 1)$$ and $$E' : y^2 = x(x+1)(x-1/2).$$

Note that if we replace $x$ by $2x$ and $y$ by $2y$ in the equation for $E$ we obtain an isomorphic elliptic curve $E''$ with Weierstrass equation $E: y^2 = 2x(x+1)(x - 1/2)$. Using the terminology in the literature this makes $E$ the "quadratic twist of $E'$ by $2$".

Indeed if $\sqrt{2}$ were in your field (it's not) then you could say something about the relationship between the rational points.

In fact, the reason you are struggling to find any rational points is because there are no others! Yours is the elliptic curve with Cremona label $96a1$ and you can find all you want to about it in the LMFDB here.