Positive rank quadratic twists of four elliptic curves

Let K   be a number field and Ei/KEi/K an elliptic curve defined over K   for i=1,2,3,4i=1,2,3,4. We prove that there exists a number field L containing K   such that there are infinitely many dk∈L×/(L×)2dk∈L×/(L×)2 such that Eidk(L) has positive rank, equivalently all four elliptic curves EiEi have growth of the rank over each of quadratic extensions Lk:=L(dk), more strongly, for any i1,i2,…,imi1,i2,…,im,rank(Ei(Li1⋯Lim))>rank(Ei(Li1⋯Lim−1))>⋯>rank(Ei(Li1))>rank(Ei(L)).rank(Ei(Li1⋯Lim))>rank(Ei(Li1⋯Lim−1))>⋯>rank(Ei(Li1))>rank(Ei(L)). We also prove that if each elliptic curve EiEi for i=1,2,3i=1,2,3 can be written in Legendre form over a cubic extension K of a number field k  , then there are infinitely many d∈k×/(k×)2d∈k×/(k×)2 such that Eid(K) for i=1,2,3i=1,2,3 is of positive rank.