Elliptic curves with Galois-stable cyclic subgroups of order 4

Infinitely many elliptic curves over $$\mathbf{Q }$$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect their 2. Let $$N_j(X)$$ denote the number with at least j pairs 4, and height most X. In this article we show that $$N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$$ . We also show, as $$X\rightarrow \infty $$ , $$N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})$$ precise nature error term being related to prime theorem zeros Riemann zeta-function critical strip. Here, $$c_{1,1}= 0.95740\ldots $$c_{1,2}=- 0.87125\ldots $$c_{2,1}= 0.035515\ldots are calculable constants. Lastly, no curve Q has more than 2