Let $E:y^2=x^3+ax+b$ be a given elliptic curve over $\mathbb{Q}$, and let $D$ be an integer. The quadratic twist of $E$, denoted by $E_D$, is given by the equation $E_D:Dy^2=x^3+ax+b$. A natural question arises: what can be said about the upper bound of $rank(E_D/\mathbb{Q})$? Is it difficult to determine the upper bound of the rank of the quadratic twist of a given elliptic curve?
For instance, when $b=0$ and the 2-torsion points of $E$ are ${(0,0),\infty}$, a result found in Silverman's famous book 'The Arithmetic of Elliptic Curves' states that $rank(E_D/\mathbb{Q})\le 2\omega(2D)-1$, where $\omega(2D)$ denotes the number of prime factors of $2D$.