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For an elliptic curve E defined over a number field K and 
$L/K$ a Galois extension, we study the possibilities of the Galois group Gal
$(L/K)$, when the Mordell–Weil rank of 
$E(L)$ increases from that of 
$E(K)$ by a small amount (namely 1, 2, and 3). In relation with the vanishing of corresponding L-functions at 
$s=1$, we prove several elliptic analogues of classical theorems related to Artin’s holomorphy conjecture. We then apply these to study the analytic minimal subfield, first introduced by Akbary and Murty, for the case when order of vanishing is 2. We also investigate how the order of vanishing changes as rank increases by 1 and vice versa, generalizing a theorem of Kolyvagin.
