= Inflection point of a plane cubic
= Plane cubic flex
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= Inflection points of a plane cubic
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A flex of a <smooth plane cubic> is a <point> whose tangent line has <intersection multiplicity> three with the cubic. Over an <algebraically closed field>, choose a flex $O$ as the <elliptic curve> identity. A line cuts out a <divisor> <linearly equivalent> to $3O$, so $P$ is a flex exactly when $3P=O$: the tangent <divisor> is then $3P$. Thus the flexes are precisely the geometric <3-torsion> <points>.
If the <characteristic> is not three, <prime-to-characteristic geometric torsion> gives nine distinct flexes. In <characteristic> three, the <scheme-theoretic fibre> $E[3]$ still has length nine, but need not have nine distinct <points>: an ordinary <elliptic curve> has three geometric <3-torsion> <points>, and a supersingular <elliptic curve> has only the identity. A flex chosen over the <algebraic closure> need not be rational over the original <field>.
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