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How would one put the elliptic curve $$Y^2 = a_3 X^3 + a_2 X^2 + a_1 X + a_0$$ into the Weierstrass form: $$y^2 = 4x^3 - g_2x - g_3,$$ that is, what change of variables will one need to use?

Update: It looks like $$X = x - \frac{a_2}{3a_3}, \quad Y = \frac{y}{2}$$ works.

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  • $\begingroup$ $X=x-a_3/12$, $Y=y$. $\endgroup$ Commented Jul 29, 2017 at 4:00

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First put $X=(a_3/4)X_1$ and $Y=(a_3^2/16) Y_1$. You get an equation of the form $$Y_1^2=4X_1^3+b_1X_1^2+b_2X_1+b_3$$ Now let $Y_1=y$ and $X_1=x-b_1/12$ (as per Felix Klein's suggestion).

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