From $y^2=x^3+Ax^2+Bx$ to $y^2+(1-c)xy-by=x^3-bx^2$

Question

I have two question

1. How can I transfer with a change of coordinates from $$y^2=x^3+Ax^2+Bx$$ to $$y^2+(1-c)xy-by=x^3-bx^2?$$
2. In a note of Prof. Lozano "Elliptic Curves, Modular Forms and their L-functions" I saw that some Parametrization of torsion structures so that I don't know that cases are in form " iff " or no.

For example can we say that torsion group is $\mathbb{Z}/2\mathbb{Z}$ if and only if $a^2b^2-4b^3 \neq 0$ in the $y^2=x^3+ax^2+bx?$

Answer 1

1. Yes, $y^2=x^3+ax^2+bx$ has torsion group $\mathbb{Z}/2\mathbb{Z}$ iff $a^2b^2−4b^3 \not= 0$

2. As noted in Appendix E of "Elliptic Curves, Modular Forms and their L-functions" by Álvaro Lozano-Robledo, the curves $y^2+(1−c)xy−by=x^3−bx^2$ all have a torsion group of size atleast $4$. Hence, in general the curves are not isomorphic.

Answer 2

No, it is not true that $y^2=x^3+ax^2+bx$ has torsion group isomorphic to $\mathbb{Z}/2\mathbb{Z}$ if and only if $a^2b^2-4b^3\neq 0$. For instance, consider $y^2=x^3+3x^2+2x$. Then $a^2b^2-4b^3=36-32=4\neq 0$, but the torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$.

The 2-torsion points on $y^2=x^3+ax^2+bx$ are those $(c,0)$ where $c$ is a root of $x^3+ax^2+bx$. So, if $a^2-4b$ is a square, then the $2$-torsion subgroup will be $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, instead of just $\mathbb{Z}/2\mathbb{Z}$. This is just for 2-torsion points... the curve could have other points of odd order in addition to 2-torsion points. The condition $a^2b^2-4b^3\neq 0$ simply guarantees that the curve is non-singular, so that a curve $y^2=x^3+ax^2+bx$ with $a^2b^2-4b^3\neq 0$ is an elliptic curve with one $2$-torsion point.

The parametrizations in my book describe elliptic curves with a subgroup $G$ contained in their torsion group, they do not describe elliptic curves with a specific torsion group. For instance, the curves $y^2=x^3+ax^2+bx$ with $a^2b^2-4b^3\neq 0$ are those such that there is a subgroup $G\cong \mathbb{Z}/2\mathbb{Z}$ contained in the torsion subgroup. See Examples E.1.1 and E.1.2 in the book for further clarification.