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We take $d\in {\mathbb{Q}}^{\times }$, as required for this rational elliptic curve. The tangent at $O=(1:-1:0)$ is $X+Y=0$. Its intersection with the curve has $d{Z}^3=0$, so it meets the curve three times at $O$: the identity is an inflection point of a plane cubic.

For $P,Q$, let their chord meet the diagonal cubic elliptic curve at the third point $R$, using the tangent if $P=Q$ and counting intersection multiplicities. Then join $R$ to $O$ and take the third intersection $S$. Define $P+Q=S$. Equivalently, three collinear intersection points satisfy $P+Q+R=O$.

Here inversion has the particularly simple formIndeed $P$, its displayed inverse, and $O$ lie on the same line, since swapping $X,Y$ preserves both the curve and $X+Y$. Thus addition is the chord or tangent operation followed by swapping the first two coordinates. The rule includes limiting cases involving $O$ and repeated intersections. Its associativity follows from the elliptic curve group law from Riemann-Roch: the identification $P\mapsto [P-O]$ with the degree-zero Picard group turns the line-intersection Weil divisor relation into addition of divisor classes. Since a line cuts a Weil divisor linearly equivalent to $3O$, that relation is exactly the chord rule just described.