On: New elliptic curve breaks 18-year-old record

[wbl]

In this case there is a negation missing. If a line intersects three points we have A+B+C=0. To get the group law you have to negate a point.

[intuitionist → wbl]

Of course for this to make sense you have to have a notion of 0, which is traditionally taken to be the point at infinity (so negation is negating the y-coordinate). It’s been a while since my algebraic geometry classes but IIRC this is just a useful convention.