Let $E/\mathbb{Q}$ be an elliptic curve. Its rank, denoted by $\text{rank}(E/\mathbb{Q})$, is a mysterious object. However, there exists a well-known very rough embedding $$ E(\mathbb{Q})/2\mathbb{Q} \to \text{Sel}^2(E/\mathbb{Q}) \subset \mathbb{Q}(S,2)^2, $$ where $$ \mathbb{Q}(S,2) \stackrel{\mathrm{def}}{=} \left\{ x \in \mathbb{Q}^{\times} / (\mathbb{Q}^{\times})^2 \mid v(x) \equiv 0 \mod 2, \forall v \notin S \right\}. $$ is a Selmer group of a field.
From this, we obtain the rough estimate $$ \text{rank}(E/\mathbb{Q}) \le 2\omega(\Delta_E) + 2, $$ where $\omega(\Delta_E)$ denotes the number of prime factors of the discriminant $\Delta_E$ of $E$.
In the case $E(\Bbb{Q})[2]=4$, we gain $rank(E/\Bbb{Q})\le 2ω(Δ_E)$. can we improve this inequality that bounds $rank(E_D/\Bbb{Q})$ using $D$ ?
For a certain $E/\Bbb{Q}$, I know we can discrease the left hand side by some constant by checking its homogeneous space. But is it impossible to descrease the left hand side by more drastic estimate, like bonding by $ω(Δ_E)$ for certain type of elliptic curves ?