with point at infinity $O = [0:1:0]$. Write affine points as $(x,y) = [x:y:1]$, and for an affine point $P=(x,y) \in E(\mathbb{Q})$ define $-P := (x,-y)$. Then, with respect to the chord-and-tangent group law on $E(\mathbb{Q})$,
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\begin{align*}
P + (-P) = O.
\end{align*}
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Moreover, if $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ are affine points of $E(\mathbb{Q})$ such that $x_1=x_2$ and $P \neq Q$, then $Q=-P$.