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I'm studying about elliptic curves. I'm reading Silverman's book, The Arithmetic of Elliptic Curves. I already asked things here and I was able to advance a little. I read in chapter III that a Weierstrass equation, through variable changes can be written in the form:

$y^{2}=4x^{3}+b_{2}x^{2}+2b_{4}x+b_{6}$

But a friend showed me Silverman's book Rational Points on Elliptic Curves, where it is written in session 1.3 that:

"we will restrict attention to cubics that are given in Weierstrass form, which classically consists of equations that look like

$y^{2} = 4x^{3} − g_{2}x − g_{3}.$

We will also use the slightly modified and more general equation

$y^{2} = x^{3} + ax^{2} + bx + c$,

and we will call either of them Weierstrass form."

My question is: is it possible to go from the equation

$y^{2} = x^{3} + ax^{2} + bx + c$

for the equation

$y^{2} = 4x^{3} − g_{2}x − g_{3}.$ ?

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  • $\begingroup$ It looks like the generalized form basically removes the constants, so I wouldn't be surprised if you could derive the general form by taking a derivative of the first. $\endgroup$ – user304051 Dec 18 '16 at 15:24
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    $\begingroup$ Do you know how to "depress" a cubic polynomial? (It's the cubic analog of "completing the square".) $\endgroup$ – J. M. is a poor mathematician Dec 18 '16 at 15:31
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    $\begingroup$ J.M. got it right. A linear substitution will remove the quadratic term. And another linear substitution will replace the coefficient $4$ with a $1$. $\endgroup$ – Jyrki Lahtonen Dec 18 '16 at 16:33

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