= Torsion multiples in a two-parameter Weierstrass family
{title2=$y^2+axy+by=x^3+bx^2,\quad P=(0,0)$}
Assume the <elliptic-curve discriminant> is nonzero. Then $b\ne0$, and tangency at $P$ gives $2P=(-b,b(a-1))$ and $-2P=(-b,0)$. The line through $P,2P$ has slope $1-a$ and gives $3P=(1-a,a-b-1)$, $-3P=(1-a,(1-a)^2)$. Consequently $P$ has exact order four when $a=1$, exact order five when $a=b+1$, and exact order six when $b=(a-1)(2-a)$. The nonsingularity condition excludes the zero values of $b$ that would otherwise invalidate these torsion conclusions.
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