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Does the order of the curve and the order of generator should be coprime for an elliptic curve defined over a prime field?

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The order of any point is a divisor of the curve group order, hence they are never coprime, unless your "generator" is the point at infinity.

This follows from
Lagrange's theorem: If $H$ is a subgroup of a finite group $G$, then $\lvert H\rvert$ divides $\lvert G\rvert$.

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