3-torsion ($E[3]$)

The 3-torsion of an elliptic curve $E$ is the kernel of multiplication by three:with the field of points specified. If the characteristic is not three, the geometric group over an algebraic closure is $(\mathbb{Z}/3\mathbb{Z}{)}^2$ and has nine points. The subgroup of points rational over the original field can be smaller.

For $E:y^2=x^3+1$ over $\mathbb{Q}$, the point $P=(0,1)$ has order three. The tangent slope is $3x^2/(2y)=0$, so the elliptic-curve doubling formulas give $2P=(0,-1)=-P$ and hence $3P=O$. This displays a nonzero rational 3-torsion point rather than only asserting its possible existence.

In characteristic three, the geometric-point count need not be nine. The scheme-theoretic multiplication kernel still carries degree information not captured by counting its ordinary points. Thus the characteristic restriction is essential when identifying geometric 3-torsion with two cyclic groups.