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The nonzero elliptic-curve discriminant forces $b\ne 0$. At $P$, the derivative with respect to $y$ is $b$ and that with respect to $x$ is zero, so the tangent is $y=0$. Substitution gives $x^2(x+b)=0$: its third point is $(-b,0)$. Applying the negation formula givesThe line through $P$ and $2P$ has slope $1-a$, because their x-coordinates are distinct. Substitution of $y=(1-a)x$ givesIts third point is $(1-a,(1-a{)}^2)$, interpreted with multiplicity if it coincides with a previously used point. Negating it yieldsBy associativity, $5P=O$ exactly when $3P=-2P$. Equality of their x-coordinates forces $1-a=-b$, and when $a=b+1$ their y-coordinates also agree and are zero. ThereforeSince $P\ne O$, its order in this case is exactly five. These are the torsion multiples in a two-parameter Weierstrass family formulas.