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 ?