We prove an analogue of Artin’s primitive root conjecture for two-dimensional tori ResK/Q Gm under the Generalized Riemann Hypothesis, where K is an imaginary quadratic field. As a consequence, we are able to derive a precise density formula for a given elliptic curve E over a finite prime field. One adjoins coordinates of all `-torsion points to the base field and asks for the density of the rational primes ` for which the resulting Galois extension over the base field has degree ` 2 − 1. It turns out that the density in question is essentially independent of the curves, and unless in certain special cases, even independent of the characteristic p if p 6≡ 1 (mod 4).
