# Elliptic curves of high rank over Quadratic extensions

Are there examples of elliptic curves which has rank 0 over $$\mathbb{Q}$$, but acquires a high rank ( $$\geq 2$$) over some quadratic extension?

More generally, are there known bounds for a given extension of degree $$n$$, how big can the rank be if you start with a rank 0 curve ( over $$\mathbb{Q}$$ )?

• The first question is easy. Take a curve $A$ over $\mathbb{Q}$ with rank as large as it can get. There exists (plenty of) quadratic twists $E$ of rank $0$. Mar 25 at 19:52