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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 ?

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  • $\begingroup$ A naive guess would be that the second equation is more suited for studying its reduction mod $2$. $\endgroup$
    – Jef
    Commented Feb 25, 2020 at 23:33
  • $\begingroup$ The paper is there, he is proving that the curve has stable reduction (ie. good or multiplicative) at every prime which is why he needs a model at $2$. Then he is looking at the properties of the morphism $X_0(N)\to E$ assuming the curve is modular. $\endgroup$
    – reuns
    Commented Feb 26, 2020 at 6:02
  • $\begingroup$ If you want to understand in general how one obtains the "minimal Weierstrass equation" you should look up Tate's algorithm (eg in Silverman 2, chapter IV.9). $\endgroup$
    – Chris Wuthrich
    Commented Feb 26, 2020 at 9:02
  • $\begingroup$ @reuns .@chris withrich so i undestand that when the curve has bad reduction at some point in this we should made miminal model $\endgroup$
    – Aster Phoenix
    Commented Feb 26, 2020 at 18:34

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