Torsion multiples in a two-parameter Weierstrass family ($y^2+axy+by=x^3+b{x}^2,P=(0,0)$)

Assume the elliptic-curve discriminant is nonzero. Then $b\ne 0$, 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.