Converting equation into Weierstrass form

Question

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

Answer 1

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 $$u^3+\frac{23}{48}u+\frac{181}{864},$$ if my calculations are OK. Anyway that's the idea to finish from where you are.

Answer 2

The long Weierstrass form $y^2+xy+y=x^3$ is transformed into the short Weierstrass form, namely to $$ y^2=x^3+621x+9774. $$ The formulas for the necessary substitutions are given here.