Elliptic curves with Galois-stable cyclic subgroups of order 4

Infinitely many elliptic curves over 𝐐 have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let N_j(X) denote the number of elliptic curves over 𝐐 with at least j pairs of Galois-stable cyclic subgroups of order 4, and height at most X . In this article we show that N_1(X) = c_1,1X^1/3+c_1,2X^1/6+O(X^0.105) . We also show, as X→∞ , that N_2(X)=c_2,1X^1/6+o(X^1/12) , the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann zeta-function in the critical strip. Here, c_1,1= 0.95740… , c_1,2=- 0.87125… , and c_2,1= 0.035515… are calculable constants. Lastly, we show no elliptic curve over Q has more than 2 pairs of Galois-stable cyclic subgroups of order 4.