= Rational torsion in the family x times x plus one times x plus m squared
{title2=$E(\mathbb Q)_{\rm tors}\cong\mathbb Z/2\mathbb Z\times\mathbb Z/4\mathbb Z$}
For integral $m\geq2$, suppose $E:y^2=x(x+1)(x+m^2)$ has <good reduction> at five or seven. At five, $m^2\equiv4$; at seven, $m^2\equiv2$ 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.
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