Do the equations have a solution?

[evinda]

Hello! (Smile)

I want to find the rank of an elliptic curve.

To do so, I have to check if the following equations have a solution in $\mathbb{Z}$ with $x \cdot y \neq 0$.

$$\\z^2=x^4-4p^2y^4\\z^2=-x^4+4p^2y^4\\z^2=2x^4-2p^2y^4\\z^2=-2x^4+2p^2y^4\\z^2=4x^4-p^2y^4\\z^2=-4x^4+p^2y^4\\z^2=px^4-4py^4\\z^2=-px^4+4py^4\\ z^2=p^2x^4-4y^4\\z^2=-p^2x^4+4y^4\\ z^2=2px^4-2py^4\\z^2=-2px^4+2py^4\\ z^2=4px^4-py^4\\ z^2=-4px^4+py^4\\ z^2=2p^2x^4-2y^4\\ z^2=-2p^2x^4+2y^4\\ z^2=4p^2x^4-y^4\\ z^2=-4p^2x^4+y^4 $$

I think that the only two equations that have a solution in $\mathbb{Z}$ with $x \cdot y \neq 0$ are these: $z^2=2px^4-2py^4$ and $z^2=-2px^4+2py^4$.Am I right? (Thinking)Is there a criterion that we could use in order to determine if there is a solution?

 

[jvicens]

Hello! I am interested in your question about finding the rank of an elliptic curve. You are on the right track by checking for solutions in $\mathbb{Z}$ with $x \cdot y \neq 0$ for each equation. However, there is a more efficient way to determine the rank of an elliptic curve.

One approach is to use the Mordell-Weil theorem, which states that the rank of an elliptic curve is equal to the number of independent rational points on the curve. In other words, if we can find a set of points on the curve that are not related by a finite translation, then the rank is equal to the number of points in that set.

Another approach is to use the Nagell-Lutz theorem, which states that the rank of an elliptic curve is equal to the number of torsion points on the curve. Torsion points are points on the curve with finite order, and they can be found by solving the curve's defining equation for $x$ and $y$ in terms of $z$. Then, by substituting in integer values for $z$, we can find the corresponding $x$ and $y$ values and determine the number of torsion points.

I hope this helps in your quest to find the rank of an elliptic curve. Good luck in your research!