Structure of torsion subgroups in an elliptic curve with integer coefficients

Question

Suppose we have an elliptic curve $E: y^{2} = (x+a)(x+b)(x+c)$ where $a,b,c \in \mathbb{Z}$. I want to know the structure of the torsion subgroup $E(\mathbb{Q})_{tors}$ based on the properties of the integers $a,b,c.$ I have seen a paper by Ken Ono titled 'Euler’s concordant forms', Acta Arithmetica, 1996, http://matwbn.icm.edu.pl/ksiazki/aa/aa78/aa7821.pdf. In this paper the author discusses about possible structures of the torsion subgroup for the elliptic curve $E_{\mathbb{Q}}(M,N) : x^{3}+(M+N)x^{2}+MNx.$ Here the torsion groups are given based on the properties of $M$ and $N$. I am looking for a link between the curves $E$ and $E_{\mathbb{Q}}(M,N).$

Answer 1

If you make the linear transform $x \mapsto x - a$, the original curve becomes

$$y^2 = x(x + b - a)(x + c - a) = x^3 + (M+N)x^2 + MNx$$

where $M = b-a$ and $N=c-a$.