Rational torsion in the family x times x plus one times x plus m squared ($E(\mathbb{Q}{)}_{\mathrm{tors}}\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$)

For integral $m\ge 2$, suppose $E:y^2=x(x+1)(x+m^2)$ has good reduction at five or seven. At five, $m^2\equiv 4$; at seven, $m^2\equiv 2$ or $4$. Each resulting reduced elliptic curve has eight points. These restrictions follow from $\Delta =16m^4(m^2-1{)}^2$ and $c_4=16(m^4-m^2+1)$: if an odd prime divides $m(m^2-1)$, the j-invariant of an elliptic curve has negative valuation. The torsion-freeness of the formal group over Qp for odd p bounds the whole rational torsion by eight. The point $(m,m(m+1))$ doubles to $(0,0)$, and $(-1,0)$ is an independent 2-torsion point. These generate the stated group of order eight.