Elliptic curves with twists of zero rank

Question

I am new to field of Elliptic curves. When I was seeing some papers related to this area I have come across elliptic curves having quadratic(some times cubic ) twists with zero rank. What is the consequence by this? Any help is great. Thank you in advance.

Answer 1

If $E/\mathbb Q$ is an elliptic curve, it is conjectured that one-half of the quadratic twists of $E$ have rank $0$, and one-half have rank $1$ ("one-half" being understood asymptotically, with twists ordered by conductor).

If $E/\mathbb Q$ is a CM curve admitting cubic (resp. quartic) twists, then it is conjectured that the proportion of cubic (resp. quartic) twists of $E$ with positive rank is $0$.

Conjecturally, what you have said should be true of any elliptic curve over $\mathbb Q$.

For reference, see this paper.