I have to convert the equation y 2 + x y + y = x 3 by a change of linear variables to the form Y 2 = X 3 + a X + b where a and b are rational numbers. So far, by completing the square method I've reduced it to Y 2 = x 3 + x 2 / 4 + x / 2 + 1 / 4where Y = y + ( x + 1 ) / 2. However, I can't figure out how to reduce it to the form asked in the question. Any help would be appreciated thanks.

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I have to convert the equation       y    2    +  x  y  +  y  =      x    3   by a change of linear variables to the form       Y    2    =      X    3    +  a  X  +  b where   a and   b are rational numbers. So far, by completing the square method I&#039;ve reduced it to      Y    2    =      x    3    +      x    2        /    4  +  x      /    2  +  1      /    4where   Y  =  y  +  (  x  +  1  )      /    2. However, I can&#039;t figure out how to reduce it to the form asked in the question. Any help would be appreciated thanks.

britspears523jp · Accepted Answer

The long Weierstrass form       y    2    +  x  y  +  y  =      x    3   is transformed into the short Weierstrass form, namely to      y    2    =      x    3    +  621  x  +  9774.

glycleWogry · Accepted Answer

Now try a substitution   x  =  u  +  k with   k a constant, multiply out and choose   k so there is no       u    2   term. With   k  =  −  1      /    12 the cubic in   x then in terms of   u is      u    3    +      23    48    u  +      181    864    ,if my calculations are OK. Anyway that&#039;s the idea to finish from where you are.

I have to convert the equation y 2 </msup> + x y + y =

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