Growth of the analytic rank of modular elliptic curves over quintic extensions

Given F a totally real number field and E/F a modular elliptic curve, we denote by G5(E/F; X) the number of quintic extensions K of F such that the norm of the relative discriminant is at most X and the analytic rank of E grows over K, i.e., ran(E/K) > ran(E/F). We show that G5(E/F; X) +∞ X when the elliptic curve E/F has odd conductor and at least one prime of multiplicative reduction. As Bhargava, Shankar and Wang [1] showed that the number of quintic extensions of F with norm of the relative discriminant at most X is asymptotic to c5,FX for some positive constant c5,F, our result exposes the growth of the analytic rank as a very common circumstance over quintic extensions.