In the paper of G.Frey there is a link between stable elliptic curves and certain Diophantine equations. The Frey curve of the equation $A-B=C$ is

$$E :\;y^2=x(x-A)(x-B)$$

where $A=a^p$, $B=b^p$, $C=c^p$.

And he define also the minimal equation of $E$ by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of $E$ becomes

$$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$$

And the discriminant is $\Delta= \frac{(ABC)^2 }{2^8}$.

My question is why Frey make this change of variable and define two equations of the curve? and why he takes specifically this change $x=4X$, $y=4X+8Y$ what is the aim of all this ?

In the paper of G.Frey there is a link between stable elliptic curves and certain Diophantine equations. The Frey curve of the equation $A-B=C$ is

$$E :\;y^2=x(x-A)(x-B)$$

where $A=a^p$, $B=b^p$, $C=c^p$.

And he define also the minimal equation of $E$ by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of $E$ becomes

$$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$$

And the discriminant is   $\Delta= \frac{(ABC)^2 }{2^8}$.

My question is why Frey make this change of variable and define two equations of the curve? and why he takes specifically this change $x=4X$, $y=4X+8Y$ what is the aim of all this ?

In the paper of G.Frey link between stable elliptic curve and certain diophantine equation .the frey-curve of the equation A-B=C is

E :$y^2=x(x-A)(x-B) $

where $A=a^p$  , $B=b^p$  , $C=c^p$

And he define also the minimal equation of E by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of E becomes

$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$

And the descriminant is $\Delta $=$ \frac{(ABC)^2 }{2^8}$

My question is why Frey make this change of varaible and define two equation of the curve  ?and why he take specifically this change $x=4X$  , $y=4X+8Y$ what is the aim of all this ?

In the paper of G.Frey there is a link between stable elliptic curves and certain Diophantine equations. The Frey curve of the equation $A-B=C$ is

$$E :\;y^2=x(x-A)(x-B)$$

where $A=a^p$, $B=b^p$, $C=c^p$.

And he define also the minimal equation of $E$ by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of $E$ becomes

$$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$$

And the discriminant is $\Delta= \frac{(ABC)^2 }{2^8}$.

My question is why Frey make this change of variable and define two equations of the curve? and why he takes specifically this change $x=4X$, $y=4X+8Y$ what is the aim of all this ?

In the paper of G.Frey link between stable elliptic curve and certain diophantine equation .the frey-curve of the equation A-B=C is

E :$y^2=x(x-A)(x-B) $ where $A=a^p$ , $B=b^p$ , $C=c^p$

And he define also the minimal equation of E by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of E becomes

$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$

And the descriminant is $\Delta $=$ \frac{(ABC)^2 }{2^8}$

My question is why Frey make this change of varaible and define two equation of the curve ?and why he take specifically this change $x=4X$ , $y=4X+8Y$ what is the aim of all this ?

In the paper of G.Frey link between stable elliptic curve and certain diophantine equation .the frey-curve of the equation A-B=C is

E :$y^2=x(x-A)(x-B) $ 

where $A=a^p$ , $B=b^p$ , $C=c^p$

And he define also the minimal equation of E by the the change of variable $x=4X$ and $y=4X+8Y$ and the the equation of E becomes

$ Y^2+XY=X^3+ \frac{A+B-1}{4}X^2+\frac{AB}{16}X$

And the descriminant is $\Delta $=$ \frac{(ABC)^2 }{2^8}$

My question is why Frey make this change of varaible and define two equation of the curve ?and why he take specifically this change $x=4X$ , $y=4X+8Y$ what is the aim of all this ?