We know from [B. Mazur, Modular curves and the Eisenstein ideal, Publ. math. IHES 47 (1977), 33-186] that if $C$ is an elliptic curve of the form ($C:y²=x³+ax+b$ with $a,b∈ℤ$), then $C(ℚ)^{tors}$ (the subgroup of elements of finite order in $C(ℚ)$ and $C(ℚ)$ is the Mordell-Weil group) is one of the following $15$ groups:
(a) $ℤ/nℤ$, with $1≤n≤10$ or $n=12$, or
(b) $ℤ/2mℤ× ℤ/2ℤ$, $1≤m≤4$.
Note that each of these groups occurs infinitely often as the torsion subgroup of an elliptic curve over $ℚ$. My question is: .What is the argument (justification) used to dsitinguish the two cases (a) and (b) (given $C/Q$, how do we tell if we are in one case of (a) or one case of (b))?
Let $E/\mathbb{Q}: y^2=x^3+ax+b$ be an elliptic curve. Then, by Mazur's theorem, $E(\mathbb{Q})_\text{tors}$ falls in case (b) if and only if the two torsion $E(\mathbb{Q})[2]\cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ is completely defined over $\mathbb{Q}$, and this occurs if and only if the polynomial $x^3+ax+b$ factors completely into (non-repeated) linear factors over $\mathbb{Q}$. Notice that in case (a) the two torsion over $\mathbb{Q}$ is either trivial of $E(\mathbb{Q})[2]\cong \mathbb{Z}/2\mathbb{Z}$.