Does an Elliptic Curve has to have a rational point by definition?

Solution 1:

You are right that an elliptic curve must always have a rational point, since at the very least you'll want to have a point that acts as the identity element.

Any elliptic curve can be put into Weiestrass form; with respect to this form, you always have the point at infinity, which is taken to be the identity element. Can you see how that is indeed a rational point?

In general, there are genus-1 curves without rational points, and a fortiori these cannot be put into Weierstrass form.

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