We establish asymptotic lower bounds for the number of elliptic curves over \( \mathbb {Q} \) with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as 1-parameter families arising from certain genus 0 modular curves. We apply techniques from the geometry of numbers and sieve methods to prove that the number of elliptic curves with unexplained entanglements \(\mathbb {Q}(E[2]) \cap \mathbb {Q}(E[3]) \ne \mathbb {Q}\) and \(\mathbb {Q}(E[2]) \cap \mathbb {Q}(E[5]) \ne \mathbb {Q}\) and naive height \(\le X\) , grows as \( \gg X^{1/9} \) and \( \gg X^{1/12} \) , respectively.
