Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ division fields $\mathbb{Q}(E[m_1]) \, \cap \, \mathbb{Q}(E[m_2])$ is non-trivial where $\gcd(m_1,m_2) = 1$.

One can show that if an elliptic curve $E$ has non-square discriminant,then $E$ will always have entanglement fields. Indeed, since $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2])$ and $\mathbb{Q}(\sqrt{\Delta_E})$ is an abelian extension, we have that $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \subseteq \mathbb{Q}(\zeta_n)$ for some $n$ by Kronecker--Weber. The Weil-pairing tells us that $\mathbb{Q}(\zeta_n) \subseteq \mathbb{Q}(E[n])$, and so $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \cap \mathbb{Q}(E[n])$. For a more precise statement see Proposition 22 of Serre's work .

I am giving a presentation on this topic and some related work, and I want to make a ``precise as possible" statement about the quantity of elliptic curves that have such entanglement. Hence, my question is as follows:

What percentage/proportion/density of elliptic curves have non-square discriminant?

Thank you in advance for your time!

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ division fields $\mathbb{Q}(E[m_1]) \, \cap \, \mathbb{Q}(E[m_2])$ is non-trivial where $\gcd(m_1,m_2) = 1$.

One can show that if an elliptic curve $E$ has non-square discriminant,then $E$ will always have entanglement fields. Indeed, since $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2])$ and $\mathbb{Q}(\sqrt{\Delta_E})$ is an abelian extension, we have that $\mathbb{Q}(\sqrt{\Delta_E})\subseteq \mathbb{Q}(\zeta_n)$ for some $n$ by Kronecker--Weber. The Weil-pairing tells us that $\mathbb{Q}(\zeta_n) \subseteq \mathbb{Q}(E[n])$, and so $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \cap \mathbb{Q}(E[n])$.

Edit As mentioned below, the above $n$ from Kronecker--Weber could in fact be even. In this case, we do not consider $E$ to have entanglement. 

For a more precise statement see Proposition 22 of Serre's work .

I am giving a presentation on this topic and some related work, and I want to make a ``precise as possible" statement about the quantity of elliptic curves that have such entanglement. Hence, my question is as follows:

What percentage/proportion/density of elliptic curves have non-square discriminant?

Thank you in advance for your time!

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ division fields $\mathbb{Q}(E[m_1]) \, \cap \, \mathbb{Q}(E[m_2])$ is non-trivial.

One can show that if an elliptic curve $E$ has non-square discriminant,then $E$ will always have entanglement fields. Indeed, since $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2])$ and $\mathbb{Q}(\sqrt{\Delta_E})$ is an abelian extension, we have that $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \subseteq \mathbb{Q}(\zeta_n)$ for some $n$ by Kronecker--Weber. The Weil-pairing tells us that $\mathbb{Q}(\zeta_n) \subseteq \mathbb{Q}(E[n])$, and so $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \cap \mathbb{Q}(E[n])$. For a more precise statement see Proposition 22 of Serre's work .

I am giving a presentation on this topic and some related work, and I want to make a ``precise as possible" statement about the quantity of elliptic curves that have such entanglement. Hence, my question is as follows:

What percentage/proportion/density of elliptic curves have square-free discriminant?

Thank you in advance for your time!

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ division fields $\mathbb{Q}(E[m_1]) \, \cap \, \mathbb{Q}(E[m_2])$ is non-trivial where $\gcd(m_1,m_2) = 1$.

One can show that if an elliptic curve $E$ has non-square discriminant,then $E$ will always have entanglement fields. Indeed, since $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2])$ and $\mathbb{Q}(\sqrt{\Delta_E})$ is an abelian extension, we have that $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \subseteq \mathbb{Q}(\zeta_n)$ for some $n$ by Kronecker--Weber. The Weil-pairing tells us that $\mathbb{Q}(\zeta_n) \subseteq \mathbb{Q}(E[n])$, and so $\mathbb{Q}(\sqrt{\Delta_E}) \subseteq \mathbb{Q}(E[2]) \cap \mathbb{Q}(E[n])$. For a more precise statement see Proposition 22 of Serre's work .

I am giving a presentation on this topic and some related work, and I want to make a ``precise as possible" statement about the quantity of elliptic curves that have such entanglement. Hence, my question is as follows:

What percentage/proportion/density of elliptic curves have non-square discriminant?

Thank you in advance for your time!